Question

Using a real life example explain where the formula for conditional probability [P(B|A) = P(A ∩B) P(A) ] comes from. Hence, or otherwise, explain why, for independent events, the probability of the two events both occurring is the same as the probability of the first event occurring multiplied by the second event occurring.

Answer 1

P(B|A) = P(A ∩B) P(A)

EXAMPLE-

In a card game, suppose a player needs to draw two cards of the same suit in order to win. Of the 52 cards, there are 13 cards in each suit. Suppose first the player draws a heart. Now the player wishes to draw a second heart. Since one heart has already been chosen, there are now 12 hearts remaining in a deck of 51 cards. So the conditional probability P(Draw second heart|First card a heart) = 12/51.

Suppose an individual applying to a college determines that he has an 80% chance of being accepted, and he knows that dormitory housing will only be provided for 60% of all of the accepted students. The chance of the student being accepted and receiving dormitory housing is defined by
P(Accepted and Dormitory Housing) = P(Dormitory Housing|Accepted)P(Accepted) = (0.60)*(0.80) = 0.48.