Introduction to Probability

How do you calculate conditional probability?

Quick answer

To calculate a conditional probability, divide the probability of both events happening together (the intersection) by the probability of the given condition: P(A|B) = P(A and B) / P(B). This determines the likelihood of an outcome after another has occurred. The given condition restricts your sample space, making P(B) your new denominator.

The topic explained

The condition shrinks your sample space

The core idea behind conditional probability is that extra information eliminates possibilities. Knowing that an event has already occurred does not change the experiment itself, but it removes everything that fails to meet that condition. Consequently, your total pool of possible outcomes becomes smaller.

Because the pool shrinks, your denominator changes from the entire sample space to just the condition. For instance, suppose a group of 20 students has 8 students taking Arabic, and 3 of those 8 also take French. If you pick a student at random and are told they take Arabic, the sample space is no longer 20. It is 8. The probability that this student takes French given that they take Arabic is 3 out of 8.

The formula: intersection divided by condition

In probability notation, the vertical bar means 'given'. The formula for P(A|B), read as 'the probability of A given B', is P(A and B) divided by P(B). You take the probability of both events occurring together and divide it by the probability of the given event.

On a two-way frequency table, this calculation is very simple. The intersection of A and B is a single cell where the two categories cross. The condition B is the total of that specific row or column. You simply divide the cell count by that row or column total.

Why the direction matters

P(A|B) is rarely equal to P(B|A). While the numerator remains the shared overlap in both cases, the denominator changes entirely based on which event comes after the word 'given'.

Consider our student example again: 8 take Arabic, 9 take French, and 3 take both. The chance a student takes French given Arabic is 3 out of 8 (0.375). But the chance a student takes Arabic given French is 3 out of 9 (around 0.333). To avoid swapping them, identify the word 'given' first and write down that condition as your denominator before doing any math.

Free printable worksheet

Conditional Probability Practice

Use these problems to practice reading conditions, shrinking the sample space, and calculating conditional probabilities. Write each answer as a fraction in simplest form.

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Conditional Probability PracticeName:

A standard six-sided die is rolled. Find the probability that the number is 6, given that the roll is an even number.
A standard six-sided die is rolled. Find the probability that the number is greater than 3, given that the roll is an odd number.
In a club of 30 members, 12 play chess, 10 play bridge, and 4 play both. Find P(Bridge | Chess).
Using the same club data (12 play chess, 10 play bridge, 4 play both), find P(Chess | Bridge).
A survey of 50 pets shows 30 dogs and 20 cats. Among the dogs, 18 have short fur. Among the cats, 14 have short fur. A pet is chosen at random and has short fur. What is the probability that it is a dog?
Two coins are tossed. Find the probability that both are heads, given that at least one coin landed on heads.
If P(A) = 0.5, P(B) = 0.4, and P(A and B) = 0.2, calculate P(B | A).

Common questions

What does the vertical bar in P(A|B) mean?

The vertical bar is read as 'given'. The expression P(A|B) means the probability that event A occurs under the condition that event B has already occurred.

Can a conditional probability ever be greater than the unconditional probability?

Yes. Because the condition restricts the sample space to a smaller group, the proportion of desired outcomes inside that group can be much higher than in the original group.

What happens if the condition has a probability of zero?

Conditional probability is undefined if the condition has a probability of zero. You cannot divide by zero, and you cannot evaluate outcomes inside an event that is impossible.

How do independent events affect conditional probability?

If two events A and B are independent, knowing that B occurred gives no new information about A. In that case, P(A|B) is simply equal to P(A).

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