What does P(A|B) mean?

The probability of A given that B has already happened — you restrict the world to just B.

What’s the formula?

P(A|B) = P(A∩B) / P(B) — the overlap divided by the probability of the "given" event.

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

No — they have different denominators and are usually different. Confusing them is a classic error.

How does it relate to independence?

If A and B are independent, P(A|B) = P(A): knowing B tells you nothing about A.

condprob() — Conditional probability

The chance of A once you know that B has already happened.

P(A|B) = P(A∩B) / P(B)

Key idea

P(A|B), read "A given B," is the chance of A once B has already happened — you shrink the world to just B. (e.g. rain given it is cloudy)
The formula divides the overlap (the intersection) by the given event: P(A|B) = P(A and B) / P(B). (e.g. 0.2 / 0.4 = 0.5)
P(A|B) and P(B|A) are usually NOT the same — swapping them is a classic error. (e.g. P(rain|clouds) is not P(clouds|rain))
If A and B are independent (one does not affect the other), conditioning changes nothing: P(A|B) = P(A). (e.g. 0.5 = 0.5)
Rearranged, this is the multiplication rule: P(A and B) = P(A|B) x P(B). (e.g. 0.5 x 0.4 = 0.2)

Definition

$P (A ∣ B) = \frac{P ( A \cap B )}{P ( B )}$

$P (A \cap B)$: chance both A and B happen
$P (B)$: chance B happens (the known part)
$P (A ∣ B)$: chance of A given B already happened

In plain words

Once you know B happened, only the B circle is left. Of that circle, the slice that also touches A is the conditional chance — the chance of A now that B is certain.

Where you'd use it

In a class, 20% like both pizza and ice cream, and 40% like ice cream. Among the ice-cream lovers, the chance they also like pizza is P(pizza|ice cream) = 0.2 / 0.4 = 50%.
A drawn card is red. The chance it is also a king is P(king|red) = (2/52) / (26/52) = 2/26 ≈ 7.7% — conditioning on "red" halves the deck.
If P(disease ∩ positive) = 0.0099 and P(positive) = 0.0594, then P(disease|positive) ≈ 17% — a positive test on a rare disease is often wrong.

Worked example 4

If P(A∩B) = 0.2 and P(B) = 0.4, then given B has happened, A happens 50% of the time: 0.2 / 0.4 = 0.5.

Frequently asked questions

What does P(A|B) mean?: The probability of A given that B has already happened — you restrict the world to just B.
What’s the formula?: P(A|B) = P(A∩B) / P(B) — the overlap divided by the probability of the "given" event.
Is P(A|B) the same as P(B|A)?: No — they have different denominators and are usually different. Confusing them is a classic error.
How does it relate to independence?: If A and B are independent, P(A|B) = P(A): knowing B tells you nothing about A.