Probability

See also: numbers, Permutations and Combinations

Conditional Probability

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

The conditional probability of A given B is defined by

P(A | B) = P(A ∩ B)
P(B)

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

If events A and B are mutually exclusive, P(A ∩ B) = 0. Hence, if P(B) > 0, then P(A | B) = 0.

Example:

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(R ∩ C) = 0.3

P(R | C) = P(R ∩ C)
P(C)
= 0.3
0.4
= 0.75

Another example:

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(V | M) = 0.3

P(V | M) = P(V ∩ M)
P(M)
0.3 = P(V ∩ M)
0.6
P(V ∩ M) = 0.3 × 0.6
P(V ∩ M) = 0.18

Eighteen percent of the city residents are vegetarian men.

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Independent and Mutually exclusive events
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Bayes' Theorem

See also: numbers, Permutations and Combinations