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• What Is Probability Sampling? | Types & Examples

What Is Probability Sampling? | Types & Examples

Published on July 5, 2022 by Kassiani Nikolopoulou . Revised on June 22, 2023.

Probability sampling is a sampling method that involves randomly selecting a sample, or a part of the population that you want to research. It is also sometimes called random sampling.

To qualify as being random, each research unit (e.g., person, business, or organization in your population) must have an equal chance of being selected. This is usually done through a random selection process, like a drawing.

Table of contents

Types of probability sampling, examples of probability sampling methods, probability vs. non-probability sampling, advantages and disadvantages of probability sampling, other interesting articles, frequently asked questions about probability sampling.

There are four commonly used types of probability sampling designs:

Simple random sampling

• Stratified sampling

Systematic sampling

• Cluster sampling

Simple random sampling gathers a random selection from the entire population, where each unit has an equal chance of selection. This is the most common way to select a random sample.

To compile a list of the units in your research population, consider using a random number generator. There are several free ones available online, such as random.org , calculator.net , and randomnumbergenerator.org .

Writing down the names of all 4,000 inhabitants by hand to randomly draw 100 of them would be impractical and time-consuming, as well as questionable for ethical reasons. Instead, you decide to use a random number generator to draw a simple random sample.

Stratified sampling collects a random selection of a sample from within certain strata, or subgroups within the population. Each subgroup is separated from the others on the basis of a common characteristic, such as gender, race, or religion. This way, you can ensure that all subgroups of a given population are adequately represented within your sample population.

For example, if you are dividing a student population by college majors, Engineering, Linguistics, and Physical Education students are three different strata within that population.

To split your population into different subgroups, first choose which characteristic you would like to divide them by. Then you can select your sample from each subgroup. You can do this in one of two ways:

• By selecting an equal number of units from each subgroup
• By selecting units from each subgroup equal to their proportion in the total population

If you take a simple random sample, children from urban areas will have a far greater chance of being selected, so the best way of getting a representative sample is to take a stratified sample.

First, you divide the population into your strata: one for children from urban areas and one for children from rural areas. Then, you take a simple random sample from each subgroup. You can use one of two options:

• Select 100 urban and 100 rural, i.e., an equal number of units
• Select 80 urban and 20 rural, which gives you a representative sample of 100 people

Systematic sampling draws a random sample from the target population by selecting units at regular intervals starting from a random point. This method is useful in situations where records of your target population already exist, such as records of an agency’s clients, enrollment lists of university students, or a company’s employment records. Any of these can be used as a sampling frame.

To start your systematic sample, you first need to divide your sampling frame into a number of segments, called intervals. You calculate these by dividing your population size by the desired sample size.

Then, from the first interval, you select one unit using simple random sampling. The selection of the next units from other intervals depends upon the position of the unit selected in the first interval.

Let’s refer back to our example about the political views of the municipality of 4,000 inhabitants. You can also draw a sample of 100 people using systematic sampling. To do so, follow these steps:

• Determine your interval: 4,000 / 100 = 40. This means that you must select 1 inhabitant from every 40 in the record.
• Using simple random sampling (e.g., a random number generator), you select 1 inhabitant.
• Let’s say you select the 11th person on the list. In every subsequent interval, you need to select the 11th person in that interval, until you have a sample of 100 people.

Cluster sampling is the process of dividing the target population into groups, called clusters. A randomly selected subsection of these groups then forms your sample. Cluster sampling is an efficient approach when you want to study large, geographically dispersed populations. It usually involves existing groups that are similar to each other in some way (e.g., classes in a school).

There are two types of cluster sampling:

• Single (or one-stage) cluster sampling, when you divide the entire population into clusters
• Multistage cluster sampling, when you divide the cluster further into more clusters, in order to narrow down the sample size

Clusters are pre-existing groups, so each high school is a cluster, and you assign a number to each one of them. Then, you use simple random sampling to further select clusters. How many clusters you select will depend on the sample size that you need.

Multi-stage sampling is a more complex form of cluster sampling, in which smaller groups are successively selected from larger populations to form the sample population used in your study.

First, you take a simple random sample of departments. Then, again using simple random sampling, you select a number of units. Based on the size of the population (i.e., how many employees work at the company) and your desired sample size, you establish that you need to include 3 units in your sample.

In stratified sampling , you divide your population in groups (strata) that share a common characteristic and then select some members from every group for your sample. In cluster sampling , you use pre-existing groups to divide your population into clusters and then include all members from randomly selected clusters for your sample.

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There are a few methods you can use to draw a random sample. Here are a few examples:

• The fishbowl draw
• A random number generator
• The random number function

Fishbowl draw

You are investigating the use of a popular portable e‐reader device among library and information science students and its effects on individual reading practices. You write the names of 25 students on pieces of paper, put them in a jar, and then, without looking, randomly select three students to be interviewed for your research.

All students have equal chances of being selected and no other consideration (such as personal preference) can influence this selection. This method is suitable when your total population is small, so writing the names or numbers of each unit on a piece of paper is feasible.

Random number generator

Suppose you are researching what people think about road safety in a specific residential area. You make a list of all the suburbs and assign a number to each one of them. Then, using an online random number generator, you select four numbers, corresponding to four suburbs, and focus on them.

This works best when you already have a list with the total population and you can easily assign every individual a number.

RAND function in Microsoft Excel

If your data are in a spreadsheet, you can also use the random number function (RAND) in Microsoft Excel to select a random sample.

Suppose you have a list of 4,000 people and you need a sample of 300. By typing in the formula =RAND() and then pressing enter, you can have Excel assign a random number to each name on the list. For this to work, make sure there are no blank rows.

This video explains how to use the RAND function.

Depending on the goals of your research study, there are two sampling methods you can use:

• Probability sampling : Sampling method that ensures that each unit in the study population has an equal chance of being selected
• Non-probability sampling : Sampling method that uses a non-random sample from the population you want to research, based on specific criteria, such as convenience

Probability sampling

In quantitative research , it is important that your sample is representative of your target population. This allows you to make strong statistical inferences based on the collected data. Having a sufficiently large random probability sample is the best guarantee that the sample will be representative and the results are generalizable and free from research biases like selection bias and sampling bias .

Non-probability sampling

Non-probability sampling designs are used in both quantitative and qualitative research when the number of units in the population is either unknown or impossible to individually identify. It is also used when you want to apply the results only to a certain subsection or organization rather than the general public. Non-probability sampling is at higher risk than probability sampling for research biases like sampling bias .

You are unlikely to be able to compile a list of every practicing organizational psychologist in the country, but you could compile a list with all the experts in your area and select a few to interview.

It’s important to be aware of the advantages and disadvantages of probability sampling, as it will help you decide if this is the right sampling method for your research design.

Advantages of probability sampling

There are two main advantages to probability sampling.

• Samples selected with this method are representative of the population at large. Due to this, inferences drawn from such samples can be generalized to the total population you are studying.
• As some statistical tests, such as multiple linear regression , t test , or ANOVA , can only be applied to a sample size large enough to approximate the true distribution of the population, using probability sampling allows you to establish correlation or cause-and-effect relationship between your variables.

Disadvantages of probability sampling

Choosing probability sampling as your sampling method comes with some challenges, too. These include the following:

• It may be difficult to access a list of the entire population, due to ethical or privacy concerns, or a full list may not exist. It can be expensive and time-consuming to compile this yourself.
• Although probability sampling reduces the risk of sampling bias , it can still occur. When your selected sample is not inclusive enough, representation of the full population is skewed .

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If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

• Student’s  t -distribution
• Normal distribution
• Null and Alternative Hypotheses
• Chi square tests
• Confidence interval
• Quartiles & Quantiles
• Data cleansing
• Reproducibility vs Replicability
• Peer review
• Prospective cohort study

Research bias

• Implicit bias
• Cognitive bias
• Placebo effect
• Hawthorne effect
• Hindsight bias
• Affect heuristic
• Social desirability bias

When your population is large in size, geographically dispersed, or difficult to contact, it’s necessary to use a sampling method .

This allows you to gather information from a smaller part of the population (i.e., the sample) and make accurate statements by using statistical analysis. A few sampling methods include simple random sampling , convenience sampling , and snowball sampling .

Stratified and cluster sampling may look similar, but bear in mind that groups created in cluster sampling are heterogeneous , so the individual characteristics in the cluster vary. In contrast, groups created in stratified sampling are homogeneous , as units share characteristics.

Relatedly, in cluster sampling you randomly select entire groups and include all units of each group in your sample. However, in stratified sampling, you select some units of all groups and include them in your sample. In this way, both methods can ensure that your sample is representative of the target population .

A sampling frame is a list of every member in the entire population . It is important that the sampling frame is as complete as possible, so that your sample accurately reflects your population.

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Home » Probability Sampling – Methods, Types and Examples

Probability Sampling – Methods, Types and Examples

Table of Contents

Probability Sampling

Definition:

Probability sampling is a method of sampling where each member of a population has a known, non-zero probability of being selected to be part of the sample. This means that each member of the population has an equal chance of being selected for the sample, and the selection of one member does not influence the selection of any other member.

This method of sampling is used in research studies and surveys where the goal is to generalize the results to the larger population. By using probability sampling, researchers can ensure that the sample is representative of the population and that the results can be generalized with a known level of confidence.

Probability Sampling Methods

Probability Sampling Methods are as follows:

Simple Random Sampling

This method involves selecting a sample of individuals from the population randomly and without any bias. Each member of the population has an equal chance of being selected.

Systematic Sampling

This method involves selecting every kth member of the population, where k is a fixed interval calculated by dividing the population size by the desired sample size.

Stratified Sampling

This method involves dividing the population into homogeneous subgroups or strata, based on some relevant characteristic, and then selecting a random sample from each stratum. This ensures that each subgroup is represented in the sample.

Cluster Sampling

This method involves dividing the population into clusters or groups, such as geographical areas or schools, and then selecting a random sample of clusters. Data is then collected from all individuals in the selected clusters.

Multi-stage Sampling

This method combines two or more sampling methods, such as cluster sampling and stratified sampling, to create a more complex sample design that is appropriate for the research question and the characteristics of the population being studied.

How to conduct Probability Sampling

To conduct probability sampling, follow these general steps:

• Define the Population: Identify the population you want to study and define its characteristics.
• Determine the Sample Size: Decide on the size of the sample you want to select from the population. This should be based on the research question and the desired level of precision.
• Choose a Sampling Method: Choose the most appropriate probability sampling method based on the research question and the characteristics of the population.
• Identify the Sampling Frame: Create a list of all the individuals or units that make up the population. This is known as the sampling frame.
• Select the Sample: Use the selected probability sampling method to randomly select individuals from the sampling frame until the desired sample size is reached.
• Conduct Data Collection: Collect data from the selected individuals using appropriate data collection methods such as surveys, interviews, or observations.
• Analyze the Data: Analyze the data collected to draw conclusions and make inferences about the population.

Examples of Probability Sampling

Here are some examples of probability sampling:

• Simple Random Sampling: Suppose you want to study the attitudes of students towards their school’s policies. You could randomly select a sample of students from the school’s enrollment list, ensuring that each student has an equal chance of being selected.
• Stratified Sampling: Suppose you want to study the average income of households in a city. You could divide the population into strata based on income levels, and then randomly select a sample from each stratum in proportion to the size of the stratum in the population.
• Systematic Sampling: Suppose you want to study the customer satisfaction of a particular store. You could select every 10th customer entering the store during a specific time period to participate in the study.
• Cluster Sampling: Suppose you want to study the prevalence of a particular disease in a region. You could randomly select several neighborhoods from the region, and then randomly select a sample of individuals from each neighborhood.
• Multi-Stage Sampling: Suppose you want to study the educational attainment of a population in a country. You could first divide the country into regions, then randomly select several regions, and finally randomly select a sample of individuals from each region.

Applications of Probability Sampling

Probability sampling has various applications in research and statistical analysis. Here are some of the main applications:

• Scientific Research : Probability sampling is commonly used in scientific research to study the characteristics of a population, such as attitudes, behaviors, and health outcomes. Researchers use probability sampling to ensure that their samples are representative of the population and the results can be generalized to the population.
• Market Research: Probability sampling is used in market research to study consumer behavior, preferences, and attitudes. Companies use probability sampling to ensure that their samples are representative of their target market, and the results can be used to inform their marketing strategies.
• Public Health: Probability sampling is used in public health research to study the prevalence of diseases, risk factors, and health outcomes in a population. Public health researchers use probability sampling to ensure that their samples are representative of the population, and the results can be used to inform public health policies and interventions.
• Political Polling: Probability sampling is used in political polling to estimate the opinions and voting behavior of a population. Pollsters use probability sampling to ensure that their samples are representative of the population, and the results can be used to predict election outcomes.
• Quality Control: Probability sampling is used in quality control to monitor and improve the quality of products and services. Quality control professionals use probability sampling to select a sample of products or services for inspection, and the results can be used to identify and correct quality issues.

When to use Probability Sampling

Here are some situations where probability sampling is particularly appropriate:

• When the research question involves estimating population parameters: If the research question involves estimating population parameters, such as the mean or proportion, then probability sampling should be used to ensure that the sample is representative of the population.
• When the population is homogeneous: If the population is homogeneous, meaning that all members have similar characteristics, then probability sampling can be used to ensure that the sample is representative of the population.
• When the population is large : If the population is large, probability sampling can be used to select a smaller, manageable sample that is still representative of the population.
• When the research is exploratory : If the research is exploratory, meaning that the research question is open-ended and the goal is to generate new ideas or hypotheses, then probability sampling can be used to ensure that the sample is diverse and representative of the population.

Purpose of Probability Sampling

The purpose of probability sampling is to obtain a sample of participants that is representative of a larger population, with a known level of accuracy or confidence. The goal is to select participants for the sample in such a way that every member of the population has an equal chance of being included in the sample.

By using probability sampling, researchers can increase the likelihood that the sample accurately represents the population, which can allow them to make inferences about the population with greater confidence. Probability sampling also reduces the likelihood of bias in the sample, which can result in more accurate and reliable research findings.

Characteristics of Probability Sampling

The main characteristics of probability sampling are as follows:

• Random selection: Probability sampling involves randomly selecting participants from the population of interest. This means that every member of the population has an equal chance of being selected for the sample.
• Known probability of selection : In probability sampling, the probability of any member of the population being selected for the sample is known and can be calculated.
• Representative sample : Probability sampling aims to obtain a sample that is representative of the larger population. This means that the sample should reflect the characteristics of the population in terms of demographics, behaviors, attitudes, and other relevant variables.
• Sampling error: Probability sampling allows researchers to estimate the amount of sampling error, which is the degree of uncertainty in the sample estimates due to chance.
• Generalizability: Probability sampling is designed to increase the generalizability of the findings from the sample to the larger population. This means that researchers can make accurate inferences about the population based on the sample data.
• Elimination of bias: Probability sampling reduces the likelihood of bias in the sample, as every member of the population has an equal chance of being selected for the sample. This helps to ensure that the sample accurately reflects the population.

Advantages of Probability Sampling

There are several advantages to using probability sampling in research:

• Reduced bias: Probability sampling reduces the likelihood of bias in the sample, as every member of the population has an equal chance of being selected for the sample. This helps to ensure that the sample accurately reflects the population.
• Known sampling error: Probability sampling allows researchers to estimate the amount of sampling error, which is the degree of uncertainty in the sample estimates due to chance.
• Statistical inferences: Probability sampling provides a solid foundation for statistical inferences about the population because the sample is selected randomly and representative of the population.
• Comparability of samples: Probability sampling also allows for the comparability of samples over time, which can be useful for tracking changes in the population over time.

Disadvantages of Probability Sampling

Some Disadvantages of Probability Sampling are as follows:

• Time-consuming and expensive : Probability sampling requires a list of the population and often involves more resources and time than other sampling methods.
• Difficult to access certain populations: In some cases, it may be difficult or impossible to access certain populations, such as those who are homeless, institutionalized, or living in remote areas. This can make it challenging to obtain a representative sample.
• Limited sample size : Probability sampling may not be practical or feasible when the population is very large or when the sample size needs to be very small.
• Potential non-response bias: Despite using a probability sample, some individuals may choose not to participate in the study, which could introduce non-response bias.
• Sampling error : While probability sampling aims to minimize sampling error, there is always the potential for chance variations in the sample that can impact the accuracy of the findings.
• Limited flexibility: Probability sampling is generally more rigid and less flexible than other types of sampling methods, which can limit the ability to make changes or adapt to unexpected circumstances.

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Decoding Probability Density Functions and Statistical Analysis Techniques

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• Understanding Probability Density Functions and Analysis Techniques

Example of Validating a Probability Density Function

Practical steps for comparison, example of estimating parameters, practical application of deriving properties, example of using estimators.

When tasked with complex statistics assignments, particularly those involving probability density functions (PDFs) and various statistical analyses, it's crucial to approach the problem methodically and with precision. These types of assignments often require a deep understanding of both theoretical concepts and practical applications. Whether your goal is to validate a function as a legitimate PDF, compare datasets to the theoretical distribution, or estimate parameters accurately, each step requires careful consideration and execution. For students aiming to solve their statistical analysis assignment effectively, a structured approach is essential. This blog aims to provide a comprehensive framework to address these challenges, ensuring that you grasp each aspect thoroughly. By breaking down the process into manageable steps, we will explore how to approach these assignments systematically, enabling you to tackle your tasks with confidence and precision.

Validating a Probability Density Function (PDF)

The first step in dealing with PDFs is to verify if a given function qualifies as a valid PDF. For a function to be considered a valid PDF, it must meet two primary criteria:

• Non-Negativity: The function must be non-negative over its entire range. This means that the function should not dip below zero, as negative values are not permissible for a PDF. For example, if you have a function defined over a certain range, ensure that it remains positive throughout this range.
• Normalization: The total area under the curve of the function must equal one. This requirement ensures that the total probability represented by the PDF is 100%. To check normalization, you would integrate the function over its entire range and confirm that the result equals one.

Consider a function that you suspect might be a valid PDF. Your task is to check both non-negativity and normalization. Begin by examining whether the function remains positive within the given range. Next, integrate the function across its domain and verify that the integral equals one. If these conditions are satisfied, the function qualifies as a valid PDF.

Analyzing Datasets in Relation to a Probability Density Function

Once you have validated a PDF, you can compare it to actual datasets to assess how well the data fits the theoretical distribution. This comparison typically involves several steps:

• Visual Comparison: Create histograms of your datasets and compare them visually to the plot of the PDF. This involves plotting the data and overlaying the PDF on the histogram to see how closely the data distribution resembles the theoretical distribution.
• Summary Statistics: Analyze summary statistics such as the mean, variance, skewness, and kurtosis of your datasets. Compare these statistics to the theoretical values expected from the PDF. A close match between the data statistics and the PDF’s theoretical values indicates a good fit.

To perform this comparison effectively, start by plotting the histograms of your datasets. For instance, if you have two datasets, compare how each aligns with the PDF. Assess the fit by examining how the histograms match the shape of the PDF. Additionally, calculate summary statistics for each dataset and compare these to the values predicted by the PDF. This helps in understanding how well the dataset conforms to the expected distribution.

Estimating Parameters from Data

Estimating parameters of a PDF from sample data involves several methods. Here are some common techniques:

• Method of Moments: This technique involves matching the moments (such as mean and variance) of the sample data with the theoretical moments of the PDF. By solving for the parameter that equates the sample moments with the theoretical moments, you can estimate the parameter values.
• Maximum Likelihood Estimation (MLE): MLE is a method where you form a likelihood function based on the PDF and sample data. You then maximize this function to find the parameter estimates. This approach is widely used due to its efficiency and accuracy.
• Order Statistics: This method involves using sample quantiles (like medians or quartiles) to estimate parameters. By solving equations derived from the theoretical cumulative distribution function (CDF), you can estimate parameters from the sample data.

Suppose you need to estimate a parameter for a given PDF using sample data. You could apply the method of moments by matching sample moments with the theoretical moments of the PDF. Alternatively, use MLE by constructing a likelihood function from the PDF and sample data, and then find the parameter estimates by maximizing this function.

Deriving Theoretical Properties

For a given Probability Density Function, calculating theoretical properties is essential to understand its behavior:

• Cumulative Distribution Function (CDF): The CDF represents the probability that a random variable takes on a value less than or equal to a specific value. It is derived by integrating the PDF. The CDF helps in understanding the probability distribution of the variable.
• Moments: Moments provide insights into the distribution’s shape and spread. The first moment is the mean, while higher-order moments include variance, skewness, and kurtosis. Calculating these moments involves integration and provides valuable information about the distribution.

To derive the CDF for a PDF, integrate the PDF over the desired range. For calculating moments, perform integrals involving powers of the variable and the PDF. These calculations provide a deeper understanding of the distribution’s characteristics, such as its center, spread, and shape.

Constructing and Using Estimators

Estimators are tools used to infer parameters from data. Here’s how you can construct and use them:

• Method of Moments Estimators: Use sample moments to estimate parameters. For example, if you are estimating a parameter aaa from sample data, match the sample mean and variance to the theoretical values derived from the PDF.
• Order Statistics Estimators: Utilize sample quantiles to estimate parameters. By solving equations that relate sample quantiles to the theoretical quantiles, you can obtain parameter estimates.

When estimating a parameter using the method of moments, calculate the sample mean and variance and equate them to the theoretical moments of the PDF. For order statistics, use sample quantiles to derive estimates of the parameters. These estimators provide practical methods for parameter estimation in real-world data analysis.

Tackling assignments involving probability density functions (PDFs) and statistical analysis requires a structured and methodical approach. By meticulously validating PDFs, you ensure that the functions you work with meet the necessary criteria of non-negativity and normalization. Comparing datasets with these validated PDFs helps you assess how well theoretical models align with real-world data. Estimating parameters from data is crucial for applying theoretical models to practical situations, using techniques like the method of moments and maximum likelihood estimation. Deriving theoretical properties such as cumulative distribution functions (CDFs) and moments provides deeper insights into the behavior and characteristics of the distribution. Constructing and using estimators effectively allows you to make informed inferences about parameters based on sample data. Mastering these techniques and applying them systematically will greatly enhance your ability to address complex statistical problems. With a solid understanding and application of these methods, you will be well-prepared to excel in your statistics assignment .

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Probability

Probability  means possibility. It is a branch of mathematics that deals with the occurrence of a random event. The value is expressed from zero to one. Probability has been introduced in Maths to predict how likely events are to happen. The meaning of probability is basically the extent to which something is likely to happen. This is the basic probability theory, which is also used in the  probability distribution , where you will learn the possibility of outcomes for a random experiment. To find the probability of a single event to occur, first, we should know the total number of possible outcomes.

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Probability Definition in Math

Probability is a measure of the likelihood of an event to occur. Many events cannot be predicted with total certainty. We can predict only the chance of an event to occur i.e., how likely they are going to happen, using it. Probability can range from 0 to 1, where 0 means the event to be an impossible one and 1 indicates a certain event. Probability for Class 10 is an important topic for the students which explains all the basic concepts of this topic. The probability of all the events in a sample space adds up to 1.

For example , when we toss a coin, either we get Head OR Tail, only two possible outcomes are possible (H, T). But when two coins are tossed then there will be four possible outcomes,  i.e {(H, H), (H, T), (T, H),  (T, T)}.

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Formula for Probability

The probability formula is defined as the possibility of an event to happen is equal to the ratio of the number of favourable outcomes and the total number of outcomes.

Sometimes students get mistaken for “favourable outcome” with “desirable outcome”. This is the basic formula. But there are some more formulas for different situations or events.

Solved Examples

1) There are 6 pillows in a bed, 3 are red, 2 are yellow and 1 is blue. What is the probability of picking a yellow pillow?

Ans: The probability is equal to the number of yellow pillows in the bed divided by the total number of pillows, i.e. 2/6 = 1/3.

2) There is a container full of coloured bottles, red, blue, green and orange. Some of the bottles are picked out and displaced. Sumit did this 1000 times and got the following results:

• No. of blue bottles picked out: 300
• No. of red bottles: 200
• No. of green bottles: 450
• No. of orange bottles: 50

a) What is the probability that Sumit will pick a green bottle?

Ans: For every 1000 bottles picked out, 450 are green.

Therefore, P(green) = 450/1000 = 0.45

b) If there are 100 bottles in the container, how many of them are likely to be green?

Ans: The experiment implies that 450 out of 1000 bottles are green.

Therefore, out of 100 bottles, 45 are green.

Probability Tree

The tree diagram helps to organize and visualize the different possible outcomes. Branches and ends of the tree are two main positions. Probability of each branch is written on the branch, whereas the ends are containing the final outcome. Tree diagrams are used to figure out when to multiply and when to add. You can see below a tree diagram for the coin:

Types of Probability

There are three major types of probabilities:

Theoretical Probability

Experimental probability, axiomatic probability.

It is based on the possible chances of something to happen. The theoretical probability is mainly based on the reasoning behind probability. For example, if a coin is tossed, the theoretical probability of getting a head will be ½.

It is based on the basis of the observations of an experiment. The experimental probability can be calculated based on the number of possible outcomes by the total number of trials. For example, if a coin is tossed 10 times and head is recorded 6 times then, the experimental probability for heads is 6/10 or, 3/5.

In axiomatic probability, a set of rules or axioms are set which applies to all types. These axioms are set by Kolmogorov and are known as Kolmogorov’s three axioms. With the axiomatic approach to probability, the chances of occurrence or non-occurrence of the events can be quantified. The axiomatic probability lesson covers this concept in detail with Kolmogorov’s three rules (axioms) along with various examples.

Conditional Probability is the likelihood of an event or outcome occurring based on the occurrence of a previous event or outcome.

Probability of an Event

Assume an event E can occur in r ways out of a sum of n probable or possible equally likely ways . Then the probability of happening of the event or its success is expressed as;

The probability that the event will not occur or known as its failure is expressed as:

P(E’) = (n-r)/n = 1-(r/n)

E’ represents that the event will not occur.

Therefore, now we can say;

P(E) + P(E’) = 1

This means that the total of all the probabilities in any random test or experiment is equal to 1.

What are Equally Likely Events?

When the events have the same theoretical probability of happening, then they are called equally likely events. The results of a sample space are called equally likely if all of them have the same probability of occurring. For example, if you throw a die, then the probability of getting 1 is 1/6. Similarly, the probability of getting all the numbers from 2,3,4,5 and 6, one at a time is 1/6. Hence, the following are some examples of equally likely events when throwing a die:

• Getting 3 and 5 on throwing a die
• Getting an even number and an odd number on a die
• Getting 1, 2 or 3 on rolling a die

are equally likely events, since the probabilities of each event are equal.

Complementary Events

The possibility that there will be only two outcomes which states that an event will occur or not. Like a person will come or not come to your house, getting a job or not getting a job, etc. are examples of complementary events. Basically, the complement of an event occurring in the exact opposite that the probability of it is not occurring. Some more examples are:

• It will rain or not rain today
• The student will pass the exam or not pass.
• You win the lottery or you don’t.

Also, read: 

• Independent Events
• Mutually Exclusive Events

Probability Theory

Probability theory had its root in the 16th century when J. Cardan, an Italian mathematician and physician, addressed the first work on the topic, The Book on Games of Chance. After its inception, the knowledge of probability has brought to the attention of great mathematicians. Thus, Probability theory is the branch of mathematics that deals with the possibility of the happening of events. Although there are many distinct probability interpretations, probability theory interprets the concept precisely by expressing it through a set of axioms or hypotheses. These hypotheses help form the probability in terms of a possibility space, which allows a measure holding values between 0 and 1. This is known as the probability measure, to a set of possible outcomes of the sample space.

Probability Density Function

The Probability Density Function (PDF) is the probability function which is represented for the density of a continuous random variable lying between a certain range of values. Probability Density Function explains the normal distribution and how mean and deviation exists. The standard normal distribution is used to create a database or statistics, which are often used in science to represent the real-valued variables, whose distribution is not known.

Probability Terms and Definition

Some of the important probability terms are discussed here:

Applications of Probability

Probability has a wide variety of applications in real life. Some of the common applications which we see in our everyday life while checking the results of the following events:

• Choosing a card from the deck of cards
• Flipping a coin
• Throwing a dice in the air
• Pulling a red ball out of a bucket of red and white balls
• Winning a lucky draw

Other Major Applications of Probability

• It is used for risk assessment and modelling in various industries
• Weather forecasting or prediction of weather changes
• Probability of a team winning in a sport based on players and strength of team
• In the share market, chances of getting the hike of share prices

Problems and Solutions on Probability

Question 1: Find the probability of ‘getting 3 on rolling a die’.

Sample Space = S = {1, 2, 3, 4, 5, 6}

Total number of outcomes = n(S) = 6

Let A be the event of getting 3.

Number of favourable outcomes = n(A) = 1

i.e. A  = {3}

Probability, P(A) = n(A)/n(S) = 1/6

Hence, P(getting 3 on rolling a die) = 1/6

Question 2: Draw a random card from a pack of cards. What is the probability that the card drawn is a face card?

A standard deck has 52 cards.

Total number of outcomes = n(S) = 52

Let E be the event of drawing a face card.

Number of favourable events = n(E) = 4 x 3 = 12 (considered Jack, Queen and King only)

Probability, P = Number of Favourable Outcomes/Total Number of Outcomes

P(E) = n(E)/n(S)

P(the card drawn is a face card) = 3/13

Question 3: A vessel contains 4 blue balls, 5 red balls and 11 white balls. If three balls are drawn from the vessel at random, what is the probability that the first ball is red, the second ball is blue, and the third ball is white?

The probability to get the first ball is red or the first event is 5/20.

Since we have drawn a ball for the first event to occur, then the number of possibilities left for the second event to occur is 20 – 1 = 19.

Hence, the probability of getting the second ball as blue or the second event is 4/19.

Again with the first and second event occurring, the number of possibilities left for the third event to occur is 19 – 1 = 18.

And the probability of the third ball is white or the third event is 11/18.

Therefore, the probability is 5/20 x 4/19 x 11/18 = 44/1368 = 0.032.

Or we can express it as: P = 3.2%.

Question 4: Two dice are rolled, find the probability that the sum is:

• less than 13

Video Lectures

Introduction.

Solving Probability Questions

Probability Important Topics

Probability Important Questions

Probability Problems

• Two dice are thrown together. Find the probability that the product of the numbers on the top of the dice is: (i) 6 (ii) 12 (iii) 7
• A bag contains 10 red, 5 blue and 7 green balls. A ball is drawn at random. Find the probability of this ball being a (i) red ball (ii) green ball (iii) not a blue ball
• All the jacks, queens and kings are removed from a deck of 52 playing cards. The remaining cards are well shuffled and then one card is drawn at random. Giving ace a value 1 similar value for other cards, find the probability that the card has a value (i) 7 (ii) greater than 7 (iii) less than 7
• A die has its six faces marked 0, 1, 1, 1, 6, 6. Two such dice are thrown together and the total score is recorded. (i) How many different scores are possible? (ii) What is the probability of getting a total of 7?

Frequently Asked Questions (FAQs) on Probability

What is probability give an example, what is the formula of probability, what are the different types of probability, what are the basic rules of probability, what is the complement rule in probability.

In probability, the complement rule states that “the sum of probabilities of an event and its complement should be equal to 1”. If A is an event, then the complement rule is given as: P(A) + P(A’) = 1.

What are the different ways to present the probability value?

The three ways to present the probability values are:

• Decimal or fraction

What does the probability of 0 represent?

The probability of 0 represents that the event will not happen or that it is an impossible event.

What is the sample space for tossing two coins?

The sample space for tossing two coins is: S = {HH, HT, TH, TT}

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There are 3 boxes Box A contains 10 bulbs out of which 4 are dead box b contains 6 bulbs out of which 1 is dead box c contains 8 bulbs out of which 3 are dead. If a dead bulb is picked at random find the probability that it is from which box?

Probability of selecting a dead bulb from the first box = (1/3) x (4/10) = 4/30 Probability of selecting a dead bulb from the second box = (1/3) x (1/6) = 1/18 Probability of selecting a dead bulb from the third box = (1/3) x (3/8) = 3/24 = 1/8 Total probability = (4/30) + (1/18) + (1/8) = (48 + 20 + 45)360 =113/360

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• Knowledge Base
• Methodology
• What Is Probability Sampling? | Types & Examples

What Is Probability Sampling? | Types & Examples

Published on 7 July 2022 by Kassiani Nikolopoulou . Revised on 7 November 2022.

Probability sampling is a sampling method that involves randomly selecting a sample, or a part of the population that you want to research. It is also sometimes called random sampling.

To qualify as being random, each research unit (e.g., person, business, or organisation in your population) must have an equal chance of being selected. This is usually done through a random selection process, like a drawing, to minimise the risk of selection bias .

Table of contents

Types of probability sampling, examples of probability sampling methods, probability vs. non-probability sampling, advantages and disadvantages of probability sampling, frequently asked questions about probability sampling.

There are four commonly used types of probability sampling designs:

Simple random sampling

Stratified sampling, systematic sampling, cluster sampling.

Simple random sampling gathers a random selection from the entire population, where each unit has an equal chance of selection. This is the most common way to select a random sample.

To compile a list of the units in your research population, consider using a random number generator. There are several free ones available online, such as random.org , calculator.net , and randomnumbergenerator.org .

Writing down the names of all 4,000 inhabitants by hand to randomly draw 100 of them would be impractical and time-consuming, as well as questionable for ethical reasons. Instead, you decide to use a random number generator to draw a simple random sample.

Stratified sampling collects a random selection of a sample from within certain strata, or subgroups within the population. Each subgroup is separated from the others on the basis of a common characteristic, such as gender, race, or religion. This way, you can ensure that all subgroups of a given population are adequately represented within your sample population.

For example, if you are dividing a student population by college majors, Engineering, Linguistics, and Physical Education students are three different strata within that population.

To split your population into different subgroups, first choose which characteristic you would like to divide them by. Then you can select your sample from each subgroup. You can do this in one of two ways:

• By selecting an equal number of units from each subgroup
• By selecting units from each subgroup equal to their proportion in the total population

If you take a simple random sample, children from urban areas will have a far greater chance of being selected, so the best way of getting a representative sample is to take a stratified sample.

First, you divide the population into your strata: one for children from urban areas and one for children from rural areas. Then, you take a simple random sample from each subgroup. You can use one of two options:

• Select 100 urban and 100 rural, i.e., an equal number of units
• Select 80 urban and 20 rural, which gives you a representative sample of 100 people

Systematic sampling draws a random sample from the target population by selecting units at regular intervals starting from a random point. This method is useful in situations where records of your target population already exist, such as records of an agency’s clients, enrollment lists of university students, or a company’s employment records. Any of these can be used as a sampling frame.

To start your systematic sample, you first need to divide your sampling frame into a number of segments, called intervals. You calculate these by dividing your population size by the desired sample size.

Then, from the first interval, you select one unit using simple random sampling. The selection of the next units from other intervals depends upon the position of the unit selected in the first interval.

Let’s refer back to our example about the political views of the municipality of 4,000 inhabitants. You can also draw a sample of 100 people using systematic sampling. To do so, follow these steps:

• Determine your interval: 4,000 / 100 = 40. This means that you must select 1 inhabitant from every 40 in the record.
• Using simple random sampling (e.g. a random number generator), you select 1 inhabitant.
• Let’s say you select the 11th person on the list. In every subsequent interval, you need to select the 11th person in that interval, until you have a sample of 100 people.

Cluster sampling is the process of dividing the target population into groups, called clusters. A randomly selected subsection of these groups then forms your sample. Cluster sampling is an efficient approach when you want to study large, geographically dispersed populations. It usually involves existing groups that are similar to each other in some way (e.g., classes in a school).

There are two types of cluster sampling:

• Single (or one-stage) cluster sampling, when you divide the entire population into clusters
• Multistage cluster sampling, when you divide the cluster further into more clusters, in order to narrow down the sample size

Clusters are pre-existing groups, so each sixth-form college is a cluster, and you assign a number to each one of them. Then, you use simple random sampling to further select clusters. How many clusters you select will depend on the sample size that you need.

Multi-stage sampling is a more complex form of cluster sampling, in which smaller groups are successively selected from larger populations to form the sample population used in your study.

First, you take a simple random sample of departments. Then, again using simple random sampling, you select a number of units. Based on the size of the population (i.e., how many employees work at the company) and your desired sample size, you establish that you need to include 3 units in your sample.

In stratified sampling , you divide your population in groups (strata) that share a common characteristic and then select some members from every group for your sample. In cluster sampling , you use pre-existing groups to divide your population into clusters and then include all members from randomly selected clusters for your sample.

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There are a few methods you can use to draw a random sample. Here are a few examples:

• The fishbowl draw
• A random number generator
• The random number function

Fishbowl draw

You are investigating the use of a popular portable e‐reader device among library and information science students and its effects on individual reading practices. You write the names of 25 students on pieces of paper, put them in a jar, and then, without looking, randomly select three students to be interviewed for your research.

All students have equal chances of being selected and no other consideration (such as personal preference) can influence this selection. This method is suitable when your total population is small, so writing the names or numbers of each unit on a piece of paper is feasible.

Random number generator

Suppose you are researching what people think about road safety in a specific residential area. You make a list of all the suburbs and assign a number to each one of them. Then, using an online random number generator, you select four numbers, corresponding to four suburbs, and focus on them.

This works best when you already have a list with the total population and you can easily assign every individual a number.

RAND function in Microsoft Excel

If your data are in a spreadsheet, you can also use the random number function (RAND) in Microsoft Excel to select a random sample.

Suppose you have a list of 4,000 people and you need a sample of 300. By typing in the formula =RAND() and then pressing enter, you can have Excel assign a random number to each name on the list. For this to work, make sure there are no blank rows.

This video explains how to use the RAND function.

Depending on the goals of your research study, there are two sampling methods you can use:

• Probability sampling : Sampling method that ensures that each unit in the study population has an equal chance of being selected
• Non-probability sampling : Sampling method that uses a non-random sample from the population you want to research, based on specific criteria, such as convenience

Probability sampling

In quantitative research , it is important that your sample is representative of your target population. This allows you to make strong statistical inferences based on the collected data. Having a sufficiently large random probability sample is the best guarantee that the sample will be representative and the results are generalisable .

Non-probability sampling

Non-probability sampling designs are used in both quantitative and qualitative research when the number of units in the population is either unknown or impossible to individually identify. It is also used when you want to apply the results only to a certain subsection or organisation rather than the general public.

You are unlikely to be able to compile a list of every practising organisational psychologist in the country, but you could compile a list with all the experts in your area and select a few to interview.

It’s important to be aware of the advantages and disadvantages of probability sampling, as it will help you decide if this is the right sampling method for your research design.

Advantages of probability sampling

There are two main advantages to probability sampling.

• Samples selected with this method are representative of the population at large. Due to this, inferences drawn from such samples can be generalised to the total population you are studying.
• As some statistical tests, such as multiple linear regression , t test , or ANOVA , can only be applied to a sample size large enough to approximate the true distribution of the population, using probability sampling allows you to establish correlation or cause-and-effect relationship between your variables.

Disadvantages of probability sampling

Choosing probability sampling as your sampling method comes with some challenges, too. These include the following:

• It may be difficult to access a list of the entire population, due to privacy concerns, or a full list may not exist. It can be expensive and time-consuming to compile this yourself.
• Although probability sampling reduces the risk of sampling bias , it can still occur. When your selected sample is not inclusive enough, representation of the full population is skewed .

When your population is large in size, geographically dispersed, or difficult to contact, it’s necessary to use a sampling method .

This allows you to gather information from a smaller part of the population, i.e. the sample, and make accurate statements by using statistical analysis. A few sampling methods include simple random sampling , convenience sampling , and snowball sampling .

Stratified and cluster sampling may look similar, but bear in mind that groups created in cluster sampling are heterogeneous , so the individual characteristics in the cluster vary. In contrast, groups created in stratified sampling are homogeneous , as units share characteristics.

Relatedly, in cluster sampling you randomly select entire groups and include all units of each group in your sample. However, in stratified sampling, you select some units of all groups and include them in your sample. In this way, both methods can ensure that your sample is representative of the target population .

A sampling frame is a list of every member in the entire population . It is important that the sampling frame is as complete as possible, so that your sample accurately reflects your population.

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7.1 - the rules of probability, example 7.1 section  .

The astragalus (ankle or heel bone) of animals were used in ancient times as a forerunner of modern dice. In fact, Egyptian tomb paintings show that sheep astragali were used in board games as early as 3500 B.C. (see Figure 7.2 ). When a sheep astragalus is thrown into the air it can land on one of four sides, which were associated with the numbers 1, 3, 4, and 6 (see Table 7.1 ). Two sides (the 3 and the 4) are wider and each come up about 40% of the time, while the narrower sides (the 1 and the 6) each come up about 10% of the time. Astragali were used for gambling, games, and divination purposes by the ancients.

  Toss an astragalus once. What's the chance you get a "1"?

Since probabilities can often be viewed as the proportion of times something happens we see our first rule of probability.

  Toss an astragalus once. What's the chance you get at least a 3?

Notice that there is another way to solve the previous problem. The opposite of "at least 3" is "getting a 1" (i.e. the only other possibility) so you can also figure the answer as 100% - 10% = 90% or 0.90. This rule of the opposites is our third rule of probability.

  Suppose you toss an astralgus twice. What's the chance that you get "4s" on both tosses?

  In ancient Rome, the lowest score in tossing four astragali (getting all four 1s) was called the dog throw. What is the probability of getting a dog throw?

Example 7.2 Independent or not? Section  

For the next two single births at Hershey Medical Center: whether the first baby is a boy and whether the second baby is a boy.

For the next two single births at Hershey Medical Center: whether the first baby is a girl and whether both babies are girls.

For the end of the month in February next year: whether there will be snow on the ground at the State College airport on February 27th and whether there will be snow on the ground at the airport on February 28th

Example 7.3 Section  

The highest paid employee has randomly selected from the list of Fortune 500 companies. Which of these probabilities is the largest?

• The chance this person is a college graduate
• The chance this person is a college graduate with a Business degree.
• The chance this person is a college graduate with an Engineering degree.

Other Interpretations Section  

While we can often think of how the process leading to data might be repeated, some events arise in situations that are not easily seen as being repeatable. In such situations, the relative frequency interpretation of probability may seem inappropriate. For example, answering a question like "What is the chance that our next President will be a woman?" would seem to require a different interpretation of the meaning of probability. Luckily, it is perfectly reasonable to assign probabilities to events outside of the relative frequency interpretation as long as they satisfy the above rules of probability. Personal probabilities that satisfy these rules give a coherent interpretation even if they might differ from one person's assignment to another's.

Visualization of basic probability assignment

• Foundation, algebraic, and analytical methods in soft computing
• Published: 19 September 2022
• Volume 26 , pages 11951–11959, ( 2022 )

Cite this article

• Hongfeng Long 1 ,
• Zhenming Peng   ORCID: orcid.org/0000-0002-4148-3331 1 &
• Yong Deng   ORCID: orcid.org/0000-0001-9286-2123 2  

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Applying geometry to the analysis and interpretation of basic probability assignment (BPA) is a unique research direction in evidence theory. Among them, the visualization of BPA helps to intuitively analyze the geometric properties and characteristics of BPA, which is an important research content in this direction, but there are currently few related studies. In response to this problem, we proposed a new BPA visualization method based on the vector representation of the BPA to illustrate the image of BPA directly. First, the basic point and the uncertain vectors could be obtained by the given BPA, and then we connected these components to construct the image of BPA. Through the image of BPA, we can effectively analyze the interaction effect of focal elements in BPA and observe the potential characteristics of BPA directly. For example, the uncertainty of focal elements can be expressed by geometric area. Moreover, the geometric meanings of parameters in the vector representation of the BPA can be explained. Finally, numerical examples are illustrated to demonstrate the advantages and related applications of the proposed method.

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Acknowledgements

The work is partially supported by National Natural Science Foundation of China (Grant Nos. 61973332 and 61775030), and is also partially supported by Natural Science Foundation of Sichuan Province of China (Grant No. 2022NSFSC40574).

The work was partially supported by National Natural Science Foundation of China (Grant No.61973332 and No.61775030).

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Hongfeng Long & Zhenming Peng

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All authors contributed to the study conception and design. HFL performed all experiments and wrote the manuscript. ZMP and YD contributed to the conception of the study and provided critical revisions. All authors read and approved the final manuscript.

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Long, H., Peng, Z. & Deng, Y. Visualization of basic probability assignment. Soft Comput 26 , 11951–11959 (2022). https://doi.org/10.1007/s00500-022-07412-1

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Accepted : 11 July 2022

Published : 19 September 2022

Issue Date : November 2022

DOI : https://doi.org/10.1007/s00500-022-07412-1

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