Conditional probability

The probability of A given B, P(A|B) = P(A ∩ B) / P(B), as restricting the sample space to B — reading it off a Venn diagram, the multiplication formula P(A ∩ B) = P(B)P(A|B), and testing independence with P(A|B) = P(A).

Statistics · Probability

Conditional probability

Why it works

Conditional probability asks: given that one event has already happened, how likely is another? Written

P(A \mid B)

— "the probability of

A

given

B

" — it measures

A

within the new, smaller world where

B

is known to have occurred.

That "smaller world" is the whole idea. Once you know

B

has happened, the only outcomes still possible are those in

B

, so

B

becomes the new sample space. The chance of

A

is then the slice of

B

that also lies in

A

P(A \mid B) = \frac{P(A \cap B)}{P(B)}.

On a Venn diagram this is simply (the overlap) ÷ (the whole of $B$ ) — you ignore everything outside

B

.For the diagram above,

P(A \mid B) = \dfrac{P(A \cap B)}{P(B)} = \dfrac{0.2}{0.2 + 0.4} = \dfrac{0.2}{0.6} = \dfrac13

.

The multiplication formula. Rearranging gives a way to find an intersection:

P(A \cap B) = P(B)\,P(A \mid B) = P(A)\,P(B \mid A).

This is the engine behind tree diagrams — multiplying along the branches.

The addition formula still holds alongside it:

P(A \cup B) = P(A) + P(B) - P(A \cap B)

.

Testing independence. Two events are independent when knowing one tells you nothing about the other — that is, the condition makes no difference:

A, B \text{ independent} \iff P(A \mid B) = P(A) \iff P(A \cap B) = P(A)\,P(B).

So to test independence, check whether

P(A \cap B)

equals

P(A)P(B)

(or whether the conditional probability equals the unconditional one). If they differ, the events are dependent — one event shifts the odds of the other.