Conditional probability

Conditional probability is the chance of an event when you already know something else has happened. Knowing more changes the chances.

P(E|F) reads 'the probability of E given F'. The formula:

P(E|F) = P(E ∩ F) / P(F), valid only when P(F) ≠ 0.

The idea: once F is known to have happened, F becomes the whole world. Out of F, how much is also E? That is the share P(E ∩ F) out of P(F).

Example: P(A) = 0.5, P(B) = 0.4, P(A ∩ B) = 0.2. Then P(A|B) = 0.2/0.4 = 0.5, and P(B|A) = 0.2/0.5 = 0.4.

So P(A|B) and P(B|A) are usually different: the chance a cricketer is left-handed is not the chance a left-handed person is a cricketer.

Properties:

• 0 ≤ P(E|F) ≤ 1.
• P(E′|F) = 1 – P(E|F): given F, either E happens or it does not.
• P(F|F) = 1. If F ⊂ E (F happening forces E), then P(E|F) = 1.
• If E and F cannot happen together, P(E|F) = 0.

Turning the formula round gives P(E ∩ F) = P(F) × P(E|F), which is the starting point of the next lessons.