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P( A ):
Probability of event A
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P( A' ):
Probability that event A does not occur
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P( B|A ):
Conditional
probability
of event B, given event A
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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.
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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.
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Probability Calculator: Solve Basic Probability Problems
The Probability Calculator computes the probability of one event, based on
probabilities of other related events.
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Specify the main goal of the analysis (via the dropdown box).
-
Enter required probabilities in the unshaded text boxes (see Hint below).
- Click the Calculate
button to create a summary report.
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This calculator handles problems that can
be addressed using three basic rules of probability - the
subtraction rule, addition
rule, and multiplication
rule.
For problems that require
Bayes' rule,
use the Bayes' Rule Calculator.
To create a report, enter data into the Probability Calculator and click the Calculate button.
Instructions: To find the answer to a frequently-asked
question, simply click on the question.
What kinds of problems can the
Probability Calculator handle?
The Probability Calculator computes an unknown probability, based
on the value of related known probabilities. Here are the types of problems
that the Probability Calculator can handle:
-
Find P(A), given P(A').
-
Find P(A), given P(B), P( B|A ) and P(A
∪
B).
-
Find P(A), given P(B), P(A
∩
B), and P(A
∪
B).
-
Find P(A), given P( B|A ) and P(A
∩
B).
-
Find P(A'), given P(A).
-
Find P(A'), given P(B), P( B|A ) and P(A
∪
B).
-
Find P(A'), given P(B), P(A
∩
B), and P(A
∪
B).
-
Find P(A'), given P( B|A ) and P(A
∩
B).
-
Find P( B|A ), given P(A) or P(A'), P(B), and P(A
∪
B).
-
Find P( B|A ), given P(A) or P(A'), and P(A
∩
B).
-
Find P(A ∪
B), given P(A) or P(A'), P(B), and P( B|A ).
-
Find P(A ∪ B), given P(A) or P(A'), P(B), and P(A
∩ B).
-
Find P(A
∩ B), given P(A) or P(A'), P(B), and P(A
∪
B).
-
Find P(A
∩ B), given P(A) or P(A'), and P( B|A).
If the above notation is confusing, see the
Notation sidebar at the top of this web page.
How can the Probability Calculator help me
solve probability problems?
Solving a probability problem is a three-step process:
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Define the problem. Specify the research goal (what you want to know), based on
the information you have.
-
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
define the problem. You select a research goal from the dropdown list box, and
you enter known probabilities into one or more text boxes.
The Probability Calculator does the rest. It applies the right
analytical technique to achieve the research goal. And it creates a summary
report that describes the analysis and presents the research finding.
What are the meanings of the various
statistical terms used by the Probability Calculator?
To use the Probability Calculator and to understand the summary
report it prepares, you need to understand some statistical jargon. If you
encounter a term that you don't understand, visit the
Statistics Glossary. All of the terms used by the Probability Calculator are
defined in the glossary.
What if I don't understand the
notation?
Refer to the Notation sidebar
at the top of this web page. All of the notation used by the Probability
Calculator is defined in the notation sidebar.
Probability Calculator
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Frequently-Asked Questions
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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 0.75. 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) = 0.75.
Now that the problem is defined, we enter the problem definition into the
Probability Calculator. Specifically, we do the following:
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Identify the main goal: Find probability of intersection of A and B.
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Set the probability of event A = 0.25.
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Set the probability of event B = 0.5.
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Set the probability of the union of events A and B = 0.75.
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) = 0.00.
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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:
-
Identify the main goal: Find probability of intersection of A and B.
-
Set the probability of event A = 0.5.
-
Set the probability of event B = 0.75.
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Set the conditional probability B, given A = 0.5.
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.
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