What is probability sampling?

• Probability 233 Definition
• Probability 233 Calculator
• Probability 233 Probability
• Probability 233 Math

Definition: Probability sampling is defined as a sampling technique in which the researcher chooses samples from a larger population using a method based on the theory of probability. For a participant to be considered as a probability sample, he/she must be selected using a random selection.

The most critical requirement of probability sampling is that everyone in your population has a known and equal chance of getting selected. For example, if you have a population of 100 people, every person would have odds of 1 in 100 for getting selected. Probability sampling gives you the best chance to create a sample that is truly representative of the population.

Probability sampling uses statistical theory to randomly select a small group of people (sample) from an existing large population and then predict that all their responses will match the overall population.

The first axiom of probability is that the probability of any event is a nonnegative real number. This means that the smallest that a probability can ever be is zero and that it cannot be infinite. The set of numbers that we may use are real numbers. Probability is traditionally considered one of the most difficult areas of mathematics, since probabilistic arguments often come up with apparently paradoxical or counterintuitive results. Examples include the Monty Hall paradox and the birthday problem.

What are the types of probability sampling?

Simple random sampling, as the name suggests, is an entirely random method of selecting the sample. This sampling method is as easy as assigning numbers to the individuals (sample) and then randomly choosing from those numbers through an automated process. Finally, the numbers that are chosen are the members that are included in the sample.

There are two ways in which researchers choose the samples in this method of sampling: The lottery system and using number generating software/ random number table. This sampling technique usually works around a large population and has its fair share of advantages and disadvantages.

Stratified random sampling involves a method where the researcher divides a more extensive population into smaller groups that usually don’t overlap but represent the entire population. While sampling, organize these groups and then draw a sample from each group separately.

• The tools of probability theory, and of the related field of statistical inference, are the keys for being able to analyze and make sense of data. These tools underlie important advances in many fields, from the basic sciences to engineering and management. This resource is a companion site to 6.041SC Probabilistic Systems Analysis and Applied Probability. It covers the same content, using.
• Probability definition is - the quality or state of being probable. How to use probability in a sentence.

A standard method is to arrange or classify by sex, age, ethnicity, and similar ways. Splitting subjects into mutually exclusive groups and then using simple random sampling to choose members from groups.

Members of these groups should be distinct so that every member of all groups get equal opportunity to be selected using simple probability. This sampling method is also called “random quota sampling.”

Random cluster samplingis a way to select participants randomly that are spread out geographically. For example, if you wanted to choose 100 participants from the entire population of the U.S., it is likely impossible to get a complete list of everyone. Instead, the researcher randomly selects areas (i.e., cities or counties) and randomly selects from within those boundaries.

Cluster sampling usually analyzes a particular population in which the sample consists of more than a few elements, for example, city, family, university, etc. Researchers then select the clusters by dividing the population into various smaller sections.Systematic sampling is when you choose every “nth” individual to be a part of the sample. For example, you can select every 5th person to be in the sample. Systematic sampling is an extended implementation of the same old probability technique in which each member of the group is selected at regular periods to form a sample. There’s an equal opportunity for every member of a population to be selected using this sampling technique.

Example of probability sampling

Let us take an example to understand this sampling technique. The population of the US alone is 330 million. It is practically impossible to send a survey to every individual to gather information. Use probability sampling to collect data, even if you collect it from a smaller population.

For example, an organization has 500,000 employees sitting at different geographic locations. The organization wishes to make certain amendments in its human resource policy, but before they roll out the change, they want to know if the employees will be happy with the change or not. However, it’s a tedious task to reach out to all 500,000 employees. This is where probability sampling comes handy. A sample from the larger population i.e., from 500,000 employees, is chosen. This sample will represent the population. Deploy a survey now to the sample.

From the responses received, management will now be able to know whether employees in that organization are happy or not about the amendment.

What are the steps involved in probability sampling?

Follow these steps to conduct probability sampling:

1. Choose your population of interest carefully: Carefully think and choose from the population, people you believe whose opinions should be collected and then include them in the sample.

2. Determine a suitable sample frame: Your frame should consist of a sample from your population of interest and no one from outside to collect accurate data.

3. Select your sample and start your survey: It can sometimes be challenging to find the right sample and determine a suitable sample frame. Even if all factors are in your favor, there still might be unforeseen issues like cost factor, quality of respondents, and quickness to respond. Getting a sample to respond to a probability survey accurately might be difficult but not impossible.

But, in most cases, drawing a probability sample will save you time, money, and a lot of frustration. You probably can’t send surveys to everyone, but you can always give everyone a chance to participate, this is what probability sample is all about.

When to use probability sampling?

Use probability sampling in these instances:

1. When you want to reduce the sampling bias: This sampling method is used when the bias has to be minimum. The selection of the sample largely determines the quality of the research’s inference. How researchers select their sample largely determines the quality of a researcher’s findings. Probability sampling leads to higher quality findings because it provides an unbiased representation of the population.

2. When the population is usually diverse: Researchers use this method extensively as it helps them create samples that fully represent the population. Say we want to find out how many people prefer medical tourism over getting treated in their own country. This sampling method will help pick samples from various socio-economic strata, background, etc. to represent the broader population.

3. To create an accurate sample: Probability sampling help researchers create accurate samples of their population. Researchers use proven statistical methods to draw a precise sample size to obtained well-defined data.

Advantages of probability sampling

Here are the advantages of probability sampling:

1. It’s Cost-effective: This process is both cost and time effective, and a larger sample can also be chosen based on numbers assigned to the samples and then choosing random numbers from the more significant sample.

2. It’s simple and straightforward: Probability sampling is an easy way of sampling as it does not involve a complicated process. It’s quick and saves time. The time saved can thus be used to analyze the data and draw conclusions.

3. It is non-technical: This method of sampling doesn’t require any technical knowledge because of its simplicity. It doesn’t require intricate expertise and is not at all lengthy.

What is the difference between probability sampling and non-probability sampling?

Here’s how you differentiate probability sampling from non-probability sampling,

The Probability Calculator

It is important to use a quality calculator if you want the calculations to be completed without any mistakes being made. This probability calculator by Calculators.tech is dependable in every manner and you can be sure that none of the results are incorrect. This is a concern for users who are calculating probability. There are various substandard calculators on the internet which should be avoided.

How to calculate probability on a calculator?

Stages of probability calculator

Here are the stages which the user has to complete to determine probability.

1. The input values which have to be entered

When the link of the calculator is opened, you would see the boxes for input values on the left. There are three input boxes and you need to enter the values for “number of possible outcomes”, “number of event occurs in A” and “number of event occurs in B”. Once these values have been entered, you can press the calculate button and advance to the next step.

2. The Output Values generated

There are 6 output values in total which are generated after the input values have been entered. These include Probability of A which is denoted by P(A). Similarly, there is P(B). The other values are A’, B’, (A ∩ B) and (A ∪ B)

Example of Probability

To understand how these values are determined, let us consider a proper example. Consider that the total number of outcomes is 10. The number of events occurred in A are 6 and The number of events occurred in B are 4. In addition to that, when these input values have been entered, you can advance to the interpretation of output values.
If the probability of A is taken as 6. The value of P (A’) would be 4. This is calculated by deducting the probability of A form the total probability which is taken as 1. Similarly, the other values would be determined by the calculator.

Main advantages of this probability calculator

There are several benefits of this calculator. Some of the key ones are listed below.

• This calculator is completely free and users do not have to make any payments for using it. A lot of tools that apparently seem free have numerous conditions attached. For instance, the tool would be free for a limited span of time. When the time span has been completed, the user would not be able to use the calculator without making payments.
• You can be absolutely sure that the data would not have any validity problems. It is dependable and you do not have to recheck anything. This calculator comes in handy for students, teachers as well as various other user types.
• This calculator is an online tool so users can use it from multiple devices. There is no need to restrict to one device and download several soft wares to use this calculator. If you want, you can add it as a widget to the website as well. This is a good alternative for users who do not want to visit the website link every now and then.
• When you talk about calculators for calculating probability or performing any other kind of calculation, the pace matters a lot. A lot of calculators are slow and users have to wait a long before results are produced. These problems are not present with this probability calculator by Calculators.tech.

How to Calculate the Probability?

The determination of probability is a stepwise process and users have to be aware of all stages. Here are all the steps which have to complete.

1. Choosing the correct event

The calculation of probability is initiated with the determination of an event. Every event has two possible outcomes. The first scenario is that it would take place and the second is that it would not. Consider that you have to determine the probability of having a Monday in one week. In other words, having Monday as the day of the week would be your event. In addition to that, the total number of days in a week is 7. Thus, the total number of outcomes would be 7.

2. Total number of outcomes

Total outcomes represent the maximum possible results that can be produced. For example, the total outcomes for a day of the week would be 7. This is simply because there are 7 days in a week.

3. Formula of Probability Calculation

The formula for calculating probability is very simple.
$\text{Probability}={\displaystyle \frac{\text{Event}}{\text{Outcomes}}}$Probability=OutcomesEvent​

To understand this formula in a better manner, we can go through another example. Consider that you have a bottle filled with 7 peanuts, 4 pistachios and 6 almonds. What is the probability that when you randomly pick one dry fruit, it would be a peanut?
We need to start by calculating the total outcomes. In this case, it would be given as
$\text{Total Outcomes}=7+4+6$Total Outcomes=7+4+6

$\text{Total Outcomes}=17$Total Outcomes=17

There are 7 peanuts in the bottle so the probability would be given as.

$\text{Probability of Peanuts}={\displaystyle \frac{7}{17}}$Probability of Peanuts=177​

$\text{Probability of Peanuts}=0.42$Probability of Peanuts=0.42

4. Total Probability should be exactly 1

When you are calculating the probability of multiple events, make sure that the total probability is 1. To elaborate on this point, we can re-consider the example given above.
In the previous step, we calculated the probability of peanuts which was 0.41. Similarly, the probability of almonds and pistachios would be given as

$\text{Probability of Pistachios}={\displaystyle \frac{4}{17}}$Probability of Pistachios=174​

$\text{Probability of Pistachios}=0.23$Probability of Pistachios=0.23

Similarly, the probability of almonds would be given as

$\text{Probability of Almonds}={\displaystyle \frac{6}{17}}$Probability of Almonds=176​

$\text{Probability of Almonds}=0.35$Probability of Almonds=0.35

Hence, the total probability would be given as

$0.35+0.23+0.42$0.35+0.23+0.42

$\text{Total probability}=1$Total probability=1

Conditional Probability

In simple terms, conditional probability refers to the occurrence of one event provided that the other has occurred. Consider that there are two events A and B. Event A occurs before event B. Hence, the conditional probability would be the probability of event B provided that event A has already occurred.

Detailed Example

Consider that there is a bad full of 6 red balls and 6 green balls. If a person takes out one red ball, it would be counted as the first event. After that, if another red ball has been taken out, the probability of this event would depend on the first event. Let us further elaborate on this example.

• When the first red ball is taken out, the probability would be 6/12.
• In the second event when conditional probability would be applied, there would be 5 red balls. Thus, the probability would be 5/11. This output would be dependent on the first red ball taken out.

Conditional probability formula

The formula of conditional probability for the events A and B would be given as

$P(A\mid B)={\displaystyle \frac{P(A\bigcap B)}{P(B)}}\text{where}P(B)>\mathrm{\Theta }$P(A∣B)=P(B)P(A⋂B)​whereP(B)>Θ

Interpretation of formula

In the above formula, conditional probability is the ratio of the probability of A intersection B and probability of B. However, an important condition in this relation is that probability of B should be greater than zero. In other cases, this formula does not hold validity.

Probability Distribution and Cumulative Probability Distribution

When you talk about probability distribution and cumulative probability distribution, they are both terms defining statistical outputs. There are obviously differences between the two terms. By going through the following points, you would be able to determine the difference between the two terms and understand the implications that each one of them has.

• Probability distribution does not involve a range of values. Instead, the possible outcomes are determined for a specific value. As it is a distribution, the results are elaborated in the form of a table. Consider that you are flipping two coins at the same time. What would be the probability that you can get a tail? The outcome would be represented by random Variable X. The possible outcomes of both coins can be
  Coin 1 is head and Coin 2 is head
  Coin 1 is head and Coin 2 is tail
  Coin 1 is tail and coin 2 is head
  Coin 1 is tail and coin 2 is tail
  If you represent the data given above in tabular form, it would be given as follows
  

  0	0.25
  1	0.5
  2	0.25

• If you have a look at the results mentioned above, the interpretation will be that there is 25% probability of getting no tails, 50% probability of getting one tail only and 25% probability of getting two tails. This is how you can determine the probability distribution.

• Cumulative probability distribution does not involve a specific value but covers a range instead. We can get more understanding if we re-consider the example mentioned above. In the case of cumulative probability, the calculation is done for a range of values. If you want to know about the chances of getting one or fewer tails, it is an example of a cumulative probability distribution. Thus, the cumulative probability would be given as
  Probability of X $\le$≤ 1 = Probability of X = 0 + Probability of X = 1.
  Considering the above example,
  $\text{Probability of}X=0is0.25$Probability of X=0is0.25
  $\text{Probability of}X=1is0.5$Probability of X=1is0.5
  Thus,
  $\text{Probability of}X\le 1=0.25+0.5$Probability of X≤1=0.25+0.5
  $\text{Probability of}X\le 1=0.75$Probability of X≤1=0.75

Difference between theoretical and experimental probability

When you talk about the difference between theoretical and experimental probability, the theoretical probability is based on expectations. It is based on estimations and assumptions. On the other hand, the experimental probability is the actual set of results produced after the calculations have been completed. Experimental probability is not based on assumptions.
Further elaboration is explained through the following points.

• Consider that you have to toss a coin for 10 times. What is the probability that you would get heads? On the basis of assumptions, you would expect that fifty percent of the outcomes would be headed. This is called theoretical probability.
• If you perform an actual experiment and toss the coin 20 times, the outcome may be different. For instance, you may get 12 heads and 8 tails. In other words, experimental probability produces actual results and no predictions or assumptions are involved in this case.

Probability 233 Definition

Union of A and B

When you talk about A and B, they are taken as two sets. Let us consider an example so that better understanding is gained.

$\text{Set A}=(2,5,6,7)$Set A=(2,5,6,7)

$\text{Set B}=(2,6,8,9)$Set B=(2,6,8,9)

The union of A and B would include all elements that are present in both sets.
If the above example is considered, the union would be given as $AUB=(2,5,6,7,8,9)$AUB=(2,5,6,7,8,9). This calculation clearly shows that all the elements of set A and B have been included in the union.

Probability 233 Calculator

It is very common to make mistakes when statistical calculations are being performed. Hence, you should use an online calculator to avoid all kinds of errors. When you talk about the union of two sets, it would include all values that are present in both sets. In other words, it would be a combination of all values. The probability distribution is related to one value carried by the variable X. The user does not have to relate the variable to any range of values.

Probability 233 Probability

Conditional probability requires a particular event to occur before the probability has been calculated. For instance, if an event A occurs, the probability that event B would occur would be determined.

FAQ's

What are the 5 rules of probability?

Answer: 5 rules are following

1. Rule 1: The probability of Any event (A) always between 0 and 1. (For any event A, 0 ≤ P(A) ≤ 1).
2. Rule 2: The sum of the probabilities of all possible outcomes is equal to 1
3. Rule 3: The Complement Rule
4. Rule 4: Addition Rule for Disjoint Events
5. Rule 5: Calculate P(A and B) using Logic

What are the 3 types of probability?

Probability 233 Math

Answer: 3 types are following

1. Classical Probability
2. Relative Frequency Definition
3. Subjective Probability

How many probability rules are there?

Answer: There are 3 basic rules for probability

How do you calculate the probability of multiple events?

Answer: Use our Calculator it is very simple and accurate