Mathématiques

What Is Conditional Probability?

How to read the vertical bar, why the denominator drops by one when a counter is not replaced, and why P(rain given cloud) is not P(cloud given rain).

Conditional probability

Conditional probability is the probability that one event happens given that another has already happened, calculated from the reduced set of outcomes still possible.

Aussi appelé: Probability given that, P(A given B)
Où les élèves le rencontrent: Higher tier GCSE and IGCSE Extended probability, usually Grade 10, in tree diagrams and Venn diagrams where an item is not replaced.

La réponse en bref
Conditional probability is the chance of one event given that another has already happened, written P(A | B) and read "the probability of A given B". Take one of 5 red counters from a bag of 10 and the next draw is 4/9 red, not 5/10.

Un exemple

P(second red | first red) = 4/9, so P(both red) = 5/10 × 4/9 = 2/9

A bag holds 5 red and 5 blue counters. The first draw is red with probability 5/10. That counter is not put back, so nine counters remain and four of them are red. The second probability is therefore conditional on the first: 4/9. Multiplying along the branches gives 20/90, which is 2/9.

Reading the notation out loud

P(A | B) is read "the probability of A given B". The vertical bar is not a division sign and not the word "and" — it separates the thing you want from the thing you already know.

Order matters, and swapping the two sides usually produces a completely different number. P(cloudy | raining) is very close to 1, because rain almost always arrives with cloud. P(raining | cloudy) is much smaller, because a great many cloudy hours pass without a drop. Same two events, same pair of words, two unrelated answers.

The denominator that drops by one

In without-replacement questions, the conditioning is done for you by the physical situation. Something has been removed, so the pool is smaller and the count of the colour you took is smaller too.

On a tree diagram, the first set of branches carries ordinary probabilities and the second set carries conditional ones. Check them by looking at the denominators: after one counter is gone from a bag of ten, every second-stage denominator is 9. Each pair of branches still adds to 1 — from a red first draw, 4/9 red and 5/9 blue.

Multiplying along a path is really the general rule P(A and B) = P(A) × P(B | A). The familiar multiply-the-probabilities rule for independent events is the special case where P(B | A) happens to equal P(B).

Shrinking the population

The other place conditional probability appears is a two-way table or Venn diagram, and there it means something simple: ignore everyone outside the condition and ask what fraction of the rest qualifies.

Of 50 students, 30 take French, 20 take German, and 12 take both. P(German | French) is 12 out of 30, or 2/5 — the 30 French students are now the whole world. P(French | German) is 12 out of 20, or 3/5. Same 12 students in the numerator, different denominators, and that is the clearest demonstration that the order of the two events cannot be swapped.

Questions fréquentes

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

It means "given". Everything to the right of the bar has already happened and is being assumed; everything to the left is what you are still asking about. It is not division, and it is not the word "and" — P(A and B) is a different quantity with a different value.

What is the formula for conditional probability?

P(A | B) = P(A and B) ÷ P(B). Reading it in words explains it: out of all the times B happens, how often does A happen too. Rearranged, it gives P(A and B) = P(B) × P(A | B), which is the rule you use when multiplying along the branches of a tree.

Is P(A | B) the same as P(B | A)?

Almost never. With 30 French students, 20 German students and 12 taking both, P(German | French) is 12/30 and P(French | German) is 12/20. The shared 12 sits over a different denominator each time. Treating the two as interchangeable is one of the most common errors in probability.

How is conditional probability shown on a tree diagram?

It is on the second set of branches. Those numbers are the probabilities for the second stage given whichever first-stage outcome you followed, which is why the two groups of second-stage branches can carry different values. Each group still adds to 1.

When does P(A | B) equal P(A)?

When the events are independent. Being told B happened leaves the probability of A exactly where it was, so the condition carries no information. Drawing with replacement is the standard classroom example: the bag is the same on the second draw, so conditioning on the first changes nothing.

Dernière mise à jour 21 août 2026