Conditional Probability

Conditioning throws away every outcome in which B did not happen and re-normalises what is left, which is why you divide by P(B). Roll one die: P(it is a 2) = 1/6, but if you are told the roll is even, only three outcomes survive and P(2 | even) = (1/6)/(1/2) = 1/3. Probabilities go in as decimals from 0 to 1 (the % unit is available too).

In September 1990 Marilyn vos Savant told Parade readers that a contestant on Let's Make a Deal should switch doors after Monty Hall opens a losing one, because switching wins 2/3 of the time. Some 10,000 letters arrived, roughly a thousand from people with PhDs, insisting she was wrong; Paul Erdős stayed unconvinced until he saw a simulation. The host's choice is not independent of where the car is, and conditioning on which door he opened is the whole game. The standard trap is confusing P(A|B) with P(B|A): almost every fatal cycling accident involves a bicycle, yet almost no bicycle ride is fatal.