Conditional probability - Higher

Conditional probability occurs when it is given that something has happened. (Hint: look for the word “given” in the question).

Example

The probability that a tennis player wins the first set of a match is \frac{3}{5}. If she wins the first set, the probability that she wins the second set is \frac{9}{10}. If she loses the first set, the probability that she wins the second set is \frac{1}{2}.

Given that the tennis player wins the second set, find the probability that she won the first set.

First, represent the information on a tree diagram:

Tree diagram showing the probability of a tennis player winning the first game in a match given they win the second.

From the tree diagram, the probability of winning the second set = \frac{27}{50} + \frac{10}{50} = \frac {37}{50}.

This means that in every 50 matches, she may win the second set 37 times (ie 37 becomes the denominator of the conditional probability). Out of those 37 times, on 27 occasions she won the first set and on 10 occasions she lost the first set.

Therefore, given that she wins the second set, the probability she won the first set is \frac{27}{37}.

There is also a formula that can be used for conditional probability:

P(A~given~B) = \frac{P(A~and~B)}{P(B)} = \frac{\frac{27}{50}}{\frac{37}{50}} = \frac{27}{37}