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Teach yourself statistics

# Probability Calculator

This Probability Calculator computes the probability of one event, based on known probabilities of other events. And it generates an easy-to-understand report that describes the analysis step-by-step.

For help in using the calculator, read the Frequently-Asked Questions or review the Sample Problems. To understand the analysis, read the Summary Report that is produced with each computation. To learn more, read Stat Trek's tutorial on the rules of probability.

• Choose the probability you want to find in the first dropdown box.
• Identify probabilities you already know in the second dropdown box.
• Enter a known probability in each text box.
• Click the Calculate button to compute a result.
• Click the Report button to describe the analysis.

## Summary Report

This calculator uses Bayes Rule (aka, Bayes theorem, the multiplication rule of probability) to compute the probability of one event, based on known probabilities of other events.

## What is Bayes Rule?

Let A be one event; and let B be any other event from the same sample space, such that P(B) > 0. Then, Bayes rule can be expressed as:

 P(A|B) = P(A) P(B|A) P(B)

where

• P(A) is the probability of Event A.
• P(B) is the probability of Event B.
• P(A|B) is the conditional probability of Event A, given Event B.
• P(B|A) is the conditional probability of Event B, given Event A.

## How to Use Bayes Rule

Bayes rule is a simple equation with just four terms. Any time that three of the four terms are known, Bayes Rule can be applied to solve for the fourth term. We've seen in the previous section how Bayes Rule can be used to solve for P(A|B). By rearranging terms, we can derive equations to solve for each of the other three terms, as shown below:

 P(B|A) = P(B) P(A|B) P(A)
 P(A) = P(B) P(A|B) P(B|A)
 P(B) = P(A) P(B|A) P(A|B)

## Extensions to Bayes Rule

The terms that are required to use Bayes Rule can be computed from other probabilities. For example,

P(A) = P(A∩B) / P(B|A)

P(A) = P(A∪B) + P(A∩B) - P(B)

P(B) = P(A∩B) / P(A|B)

P(B) = P(A∪B) + P(A∩B) - P(A)

P(A|B) = P(A∩B) / P(B)

P(B|A) = P(A∩B) / P(A)

where

• P(A∩B) is the probability of the intersection of Events A and B.
• P(A∪B) is the probability of the union of Events A and B.

Using these formulas, Bayes Rule can be rewritten through substitution to accommodate P(A∩B) and P(A∪B) as inputs. For example, here are two "new" versions of Bayes Rule:

 P(A|B) = P(A∩B) P(B)
 P(A|B) = P(A) + P(B) - P(A∪B) P(B)

To compute the probability of an event, this calculator examines the known probabilities of other events and chooses an appropriate formula to complete the computation.

Instructions: To find the answer to a frequently-asked question, simply click on the question.

### Can you explain the notation?

All of the notation used by the Probability Calculator is defined below:

• P( A ):
Probability of event A
• P( B ):
Probability of event B
• P( A|B ):
Conditional probability of event A, given event B
• P( B|A ):
Conditional probability of event B, given event A
• P(A B):
Probability that event A and/or event B occurs. This is also known as the probability of the union of A and B.
• P(A B):
Probability that event A and event B both occur. This is also known as the probability of the intersection of A and B.

### What is Bayes Rule?

Bayes Rule is an equation that expresses the conditional relationships between two events in the same sample space. Bayes Rule can be expressed as:

 P( A | B ) = P( A ) P( B | A ) P( B )

where

• P( A ) is the probability of Event A.
• P( B ) is the probability of Event B.
• P( A | B ) is the conditional probability of Event A, given Event B.
• P( B | A ) is the conditional probability of Event B, given Event A.

The probability calculator uses Bayes Rule to compute probabilities of one event, given probabilities of other related events.

### Can a computed probability be less than 0 or greater than 1.0?

If Event A occurs 100% of the time, the probability of its occurrence is 1.0; that is, P(A) = 1.0. And if Event A never occurs, the probability of its occurrence is 0. In the real world, an event cannot occur less than 0% of the time or occur more than 100% of the time; so a real-world event must have a probability between 0 and 1.0.

This calculator computes probabilities based on the inputs provided. It is possible to enter probabilities that could not occur together in the real world. When that happens, the calculator may generate a probability that could not occur in the real world; that is, the calculator could report a probability less than 0 or greater than 1.0.

To illustrate how this could happen, consider Bayes Rule:

 P( A | B ) = P( A ) P( B | A ) P( B )

where

• P(A) is the probability that Event A occurs.
• P(B) is the probability that Event B occurs.
• P(A|B) is the probability that A occurs, given that B occurs.
• P(B|A) is the probability that B occurs, given that A does not occur.

From this equation, we see that P(B) should never be less than P(A)*P(B|A); otherwise, the computed probability of P(A|B) will be greater than 1, which is not a valid outcome. For example, suppose you plug the following numbers into Bayes Rule:

• P(B) = 0.1
• P(A) = 0.5
• P(B|A) = 0.6

Given these inputs, the Probability Calculator (which uses Bayes Rule) will compute a value of 3.0 for P(A|B), clearly an invalid result. If the calculator computes a probability less than 0 or greater than 1.0, that is a warning sign. It means your probability inputs are invalid; they do not reflect real-world events.

### How can the Probability Calculator help me solve probability problems?

Solving a probability problem is a four-step process:

• Define the problem. Specify the research goal (what you want to know).
• Gather data. Collect information you need to achieve the goal.
• Analyze data. Apply the right analytical technique to achieve the research goal.
• Report results. Present the answer to the research goal.

The Probability Calculator provides a framework to help you with each critical step. From the first dropdown box, identify the probability that you wish to compute. From the second dropdown box, identify a set of probabilities that will enable you to complete the computation. Then, enter those probabilities into two or more text boxes. And finally, click the Calculate button.

The Probability Calculator does the rest. It applies the right analytical technique to the data you entered. And it creates a summary report that describes the analysis and presents the research finding.

### What is E Notation?

E notation is a way to write numbers that are too large or too small to be concisely written in a decimal format. This calculator uses E notation to express very small numbers.

With E notation, the letter E represents "times ten raised to the power of". Here is an example of a very small number written using E notation:

3.02E-12 = 3.02 * 10-12 = 0.00000000000302

If a probability can be expressed as an ordinary decimal with fewer than 14 digits, the Probability Calculator will do so. But if a probability is very small (nearly zero) and requires a longer string of digits, the calculator will use E notation to display its value.

## Sample Problems

1. Bob is running in two races - a 100-yard dash and a 200-yard dash. The probability of winning the 100-yard dash is 0.25, and the probability of winning the 200-yard dash is 0.50. The probability of winning at least one race is 0.65. What is the probability that Bob will win both races?

Solution:

The first step is to define the problem. We begin by identifying the key events:

Let event A = Bob wins the 100-yard dash.
Let event B = Bob wins the 200-yard dash.

Then, we define the main goal, in terms of these events. For the main goal, we want to know the probability of the intersection of events A and B; that is, we want to know P(A B).

Next, we specify the known probabilities:

P(A) = 0.25.
P(B) = 0.5.
P(A B) = 0.65.

Now that the problem is defined, we tun to the Probability Calculator. Specifically, we do the following:
• Select "Find P(A B)" in the first dropdown box.
• Select "P(A), P(B), and P(A B)" in the second dropdown box
• Enter 0.25 for P(A).
• Enter 0.5 for P(B).
• Enter 0.65 for P(A B).
Then, we hit the Calculate button. Calculator inputs and output are shown below. The analysis indicates that the probability that Bob will win both races is 0.10.  1. Mary is a successful pitcher for her college softball team. On average, she wins 75% of the time. However, when she gives up a home run, Mary wins only 50% of the time. She gives up a home run in half her games. In her next game, what is the probability that Mary will give up a home run and win?

Solution:

The first step is to define the problem. We begin by identifying the key events:

Let event A = Mary gives up a home run.
Let event B = Mary wins.

Then, we define the main goal, in terms of these events. For the main goal, we want to know the probability that both events occur; that is, we want to know the probability that Mary gives up a home run and Mary wins. This is the intersection of events A and B; that is, we want to know P(A B).

Next, we specify the known probabilities:

P(A) = 0.5, since Mary gives up a home run half the time.
P(B) = 0.75, since Mary wins 75% of the time.
P( B|A ) = 0.5, since Mary wins only half the time when she gives up a home run.

Now that the problem is defined, we enter the problem definition into the Probability Calculator. Specifically, we do the following:
• Select "P(A B)" in the first dropdown box.
• From the options in the second dropdown box, we select "P(A) and P(B|A)". (We select this option, because we know these probabilities.)
• Enter 0.5 for P(A).
• Enter 0.5 for P(B|A).
Then, we hit the Calculate button. This produces a summary report that describes the analytical technique and computes the probability of the intersection of events A and B. Thus, we find that P(A B) = 0.25.

Note: From the problem statement, we learned that Mary wins 75% of the time; that is, P(B) = 0.75. However, P(B) was not required to solve this problem. We only needed to know P(A) and P(B|A).

Part of the challenge in solving probability problems is distinguishing useful data from superfluous data. The Probability Calculator can help. Use the first dropdown box to choose a probability to compute. Then, use the second dropdown box to identify other probabilities that will allow you to complete the computation.

For this problem, you would select "Find P(A B)" from the first dropdown box. Then, when you look at options from the second dropdown box, you would see that one option only requires P(A) and P(B|A) to compute P(A B). ## Problem?

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