See also: numbers, Permutations and Combinations


For two events A and B, the conditional probability of event A, is the probability of event A given the occurence (or non-occurence) of event B.

The conditional probability of A given B, written as P(AB), is defined by

P ( AB ) = P ( AB ) P ( B )

If events A and B are independent, P ( AB ) = P ( A ) × P ( B ) . Hence

P ( AB ) = P ( A ) P ( BA ) = P ( B )

In other words, the probability of A given B is just the probability of A (regardless of B — whether B has occured or not doesn't affect the probability of A, because they are independent events).

Examples:

  • In a certain region, on any day of the year, the probability that it's cloudy is 0.4. It's also known that there's a 0.3 probability that any day is a cloudy and rainy day. Given that today is cloudy, what is the probability that it will rain?

    Let C be the event that it's cloudy and R be the event that it rains.

    P ( C ) = 0.4 P ( RC ) = 0.3 P ( RC ) = P ( RC ) P ( C ) = 0.3 0.4 = 0.75

    Given today is cloudy, there's a 75% chance it will rain today.

  • In a city, the ratio between men and women is 6:4. Thirty percent of the men are vegetarian. What percentage of the city residents are vegetarian men?

    Let M be the probability of any random resident being a man and V be the probability that any random resident being a vegetarian.

    P ( M ) = 0.6 P ( VM ) = 0.3 P ( VM ) = P ( VM ) P ( M ) 0.3 = P ( VM ) 0.6 P ( VM ) = 0.3×0.6 P ( VM ) = 0.18

    Eighteen percent of the city residents are vegetarian men.

By Jimmy Sie

See also: numbers, Permutations and Combinations