How to Use the Conditional Probability Calculator
The calculator helps you find conditional probability from probabilities, counts, or a 2×2 table. To use the calculator, choose the input type that matches the information given in your question and follow these steps:
- Choose Probabilities when the question gives you the joint probability, P(A ∩ B), together with P(A) or P(B).
- Select whether you want to find P(A|B) or P(B|A).
- Enter the joint probability, P(A ∩ B).
- Enter the probability of the event A or B as a decimal, depending on the probability you want to find. If you want to find P(A|B), you need to enter P(A ∩ B) and P(B). However, if you want to find P(B/A), you should enter P(A ∩ B) and P(A)
- Click Calculate
If the question provides you with frequencies or number of observations instead of probabilities, follow these steps:
- Select Counts as the data input option
- Select P(A|B) or P(B|A), depending on what you want to find
- Enter the number of observations that belong to both A and B.
- Enter the number belonging to the conditioning event.
- Click Calculate.
Tip. You do not need the total sample size unless it is needed to determine one of these counts.
Sometimes, you may be required to compute conditional probabilities from a given 2 x 2 contingency table. Our calculator still supports this option. Therefore, to find conditional probabilities from a two-way contingency table with the calculator, follow these steps:
- Select the 2×2 Table input option
- Select the probability you want to find. This option allows you to find P(A|B), P(A|not B), P(B|A), or P(B|not A).
- Enter the four cell frequencies. The calculator will automatically find the row totals, column totals, and grand total.
- Click Calculate
In each case, the calculator instantly returns the correct conditional probability and provides a clear step-by-step solution, showing you how the probabilities were computed.
What Is Conditional Probability?
Conditional probability is the probability that one event occurs when another event is already known to have occurred.
It is written using a vertical bar. For example, P(A|B) is read as “the probability of A given B.”
The word given is important. Once B is known to have occurred, we no longer consider every possible outcome. We restrict our attention to outcomes where B occurred and then determine how many of those outcomes also belong to A.
Suppose 50 students attended a review session and 40 of those students passed an exam. If we want the probability that a student passed given that the student attended the review session, only those 50 students form the relevant group. Therefore, P(Passed | Review) = 40 / 50 = 0.8. In other words, 0.8 is the conditional probability that a student passed given that the student attended the review session.
Conditional Probability Formula
The conditional probability formula of A given B is: P(A|B) = P(A ∩ B) / P(B)
where:
- P(A|B) is the conditional probability of A given B
- P(A ∩ B) is the probability that A and B both occur
- P(B) is the probability that B occurs
For the above formula to be defined, P(B) must be greater than 0.
However, if you want to reverse the condition and find the probability of B given A, the formula changes to: P(B|A) = P(A ∩ B) / P(A). For this formula to be defined, P(A) must be greater than 0.
Note. The joint probability remains the same in both formulas. Only the denominator changes.
How to Calculate Conditional Probability From Probabilities
When probabilities are given directly, identify the joint probability and the probability of the conditioning event. Then divide the joint probability by the conditioning probability.
The event that appears after the vertical bar tells you which probability belongs in the denominator.
Example 1: Find P(A|B)
A college survey found that the probability a randomly selected student uses the tutoring center and passes a statistics course is 0.28. The probability that a student uses the tutoring center is 0.40. Find the probability that a student passes given that the student uses the tutoring center.
Let:
- A = student passes statistics
- B = student uses the tutoring center
From the question, we know that:
- P(A ∩ B) = 0.28
- P(B) = 0.40
We want to find P(A|B).
Using the conditional probability formula: P(A|B) = P(A ∩ B) / P(B)
Substituting the values, we get:
P(A|B) = 0.28 / 0.40
= 0.7
Therefore, P(A|B) = 0.7. This implies that, given a student uses the tutoring center, the probability that the student passes statistics is 0.7.
Example 2: Find P(B|A)
At a university, the probability that a randomly selected student is enrolled in an online course and also works part-time is 0.24. The probability that the student is enrolled in an online course is 0.40. Find the probability that the student works part-time given that the student is enrolled in an online course.
Let:
- A = student is enrolled in an online course
- B = student works part-time
From the question:
- P(A ∩ B) = 0.24
- P(A) = 0.40
In this case, we need to find P(B|A).
Recall. P(B|A) = P(A ∩ B) / P(A)
Substituting the values, we get:
= 0.24 / 0.40
= 0.6
Therefore, P(B|A) = 0.6. This implies that given that a student is enrolled in an online course, the probability that the student works part-time is 0.6.
How to Find Conditional Probability From Counts
Sometimes a problem gives the number of observations instead of probabilities. In that case, use: P(A|B) = n(A ∩ B) / n(B)
where:
- n(A ∩ B) is the number of observations belonging to both A and B
- n(B) is the number of observations belonging to B
The same idea applies when finding P(B|A): P(B|A) = n(A ∩ B) / n(A)
The key is to divide by the number of observations in the event that comes after given.
Example 4: P(A|B) from Counts
A statistics instructor recorded that 72 students attended an exam review session. Of those students, 54 passed the exam. Find the probability that a student passed given that the student attended the review session.
Let:
- A = student passed
- B = student attended the review session
From the question:
- n(A ∩ B) = 54
- n(B) = 72
Since we are finding P(A|B), only the 72 students who attended the review session form the relevant group.
Recall. P(A|B) = n(A ∩ B) / n(B)
Substituting the values, we have:
P(A|B) = 54 / 72
= 0.75
Therefore, given that a student attended the review session, the probability that the student passed is 0.75.
Example 5: P(B|A) from Counts
A professor found that 120 students passed a final exam. Among those students, 84 had completed the optional practice problems. Find the probability that a student completed the practice problems given that the student passed.
Let:
- A = student passed the exam
- B = student completed the practice problems
From the question:
- n(A ∩ B) = 84
- n(A) = 120
We want to find P(B|A).
By definition, P(B|A) = n(A ∩ B) / n(A)
Substituting the values, we get:
P(B|A) = 84 / 120
= 0.7
Therefore, given that a student passed the exam, the probability that the student completed the practice problems is 0.7.
How to Find Conditional Probability From a 2×2 Table
A 2×2 table organizes observations according to two events. It may also be called a two-way table or contingency table.
A general 2×2 table looks like this:
| B | Not B | Total | |
|---|---|---|---|
| A | A and B | A and not B | Total A |
| Not A | not A and B | not A and not B | Total not A |
| Total | Total B | Total not B | Total |
To find conditional probability from the table:
- Look at the event after the vertical bar.
- Find the row or column total for that event.
- Use that total as the denominator.
- Find the cell where the two required events intersect.
- Divide the intersection count by the conditioning total.
This means the denominator is not automatically the grand total.
Example 7: Find P(A|B) From a Two-Way Table
A college surveyed students about whether they own a car and whether they live off campus.
Let:
- A = student owns a car
- B = student lives off campus
The results were:
| Lives Off Campus, B | Not B | Total | |
|---|---|---|---|
| Owns a Car, A | 42 | 28 | 70 |
| Not A | 18 | 62 | 80 |
| Total | 60 | 90 | 150 |
Find P(A|B).
Solution
Because B is the conditioning event, we only consider the 60 students who live off campus.
Among those students, 42 own a car.
Therefore, P(A|B) = 42 / 60
= 0.7
As such, given that a student lives off campus, the probability that the student owns a car is 0.7.
Example 8: Find P(A|not B) From a Two-Way Table
A statistics class recorded whether students attended a review session and whether they passed an exam.
Let:
- A = student passed
- B = student attended the review session
| Attended Review, B | Did Not Attend, not B | Total | |
|---|---|---|---|
| Passed, A | 48 | 36 | 84 |
| Did Not Pass, not A | 12 | 24 | 36 |
| Total | 60 | 60 | 120 |
Find P(A|not B).
Solution
The conditioning event is not B. Thus, we should focus only on students who did not attend the review session.
There are 60 students in that group, and 36 passed. Therefore, P(A|not B) = 36 / 60
= 0.6
As such, given that a student did not attend the review session, the probability that the student passed is 0.6.
Example 9: Find P(B|not A) From a Two-Way Table
A university surveyed students about whether they use the campus gym and whether they live on campus.
Let:
- A = student uses the campus gym
- B = student lives on campus
The results were:
| Lives on Campus, B | Not B | Total | |
|---|---|---|---|
| Uses Gym, A | 50 | 30 | 80 |
| Does Not Use Gym, not A | 20 | 40 | 60 |
| Total | 70 | 70 | 140 |
Find P(B|not A).
Solution
The event after the vertical bar is not A. Therefore, only the 60 students who do not use the campus gym belong to the relevant group.
Of those students, 20 live on campus.
Therefore, P(B|not A) = 20 / 60
= 0.333333
This means that given that a student does not use the campus gym, the probability that the student lives on campus is approximately 0.333333.
Conditional Probability and Independent Events
Two events are independent when knowing that one occurred does not change the probability of the other.
For independent events, P(A|B) = P(A) and P(B|A) = P(B)
For example, suppose you flip a fair coin and roll a fair die. Knowing that the coin landed heads does not change the probability of rolling a particular number on the die. Therefore, conditional probability is especially useful when events are dependent because knowing one event occurred can change the probability of the other.
Conditional Probability vs. Bayes’ Theorem
Conditional probability and Bayes’ theorem are closely related, but they are not the same calculation. The standard conditional probability formula finds P(A|B) when you already know P(A ∩ B) and P(B): P(A|B) = P(A ∩ B) / P(B)
However, Bayes’ theorem is useful when you know probabilities such as P(B|A), P(A), and P(B) and want to reverse the condition to find P(A|B).
This calculator focuses on conditional probability from direct probabilities, counts, and 2×2 tables. A Bayes’ theorem calculator is better suited to problems that require reversing a conditional probability.
If a probability problem instead asks about arrangements or selections, our Combination Calculator and Permutation Calculator can help with the counting needed to determine the number of possible outcomes.
Frequently Asked Questions
A conditional probability calculator finds the probability of one event occurring given that another event has already occurred. It can calculate values such as P(A|B) or P(B|A) from probabilities, counts, or a 2×2 table and show the calculation step by step.
Conditional probability is the probability of one event occurring given that another event has already occurred. For example, P(A|B) represents the probability of A given B.
The formula is P(A|B) = P(A ∩ B) / P(B), where P(A ∩ B) is the probability that A and B both occur and P(B) is the probability of the conditioning event.
No. They use different conditioning events and therefore usually have different denominators and different answers.
No. A valid conditional probability must fall between 0 and 1.
Divide the number of observations belonging to both events by the number belonging to the conditioning event. For P(A|B), use the formula: P(A|B) = n(A ∩ B) / n(B)
