7. Conditional Probability
If E1 and E2 are two events, the probability that E2 occurs given that E1 has occurred is denoted by P(E2|E1).
P(E2|E1) is called the conditional probability of E2 given that E1 has occurred.
Calculating Conditional Probability
Let E1 and E2 be any two events defined in a sample space S such that P(E1) > 0.
The conditional probability of E2, assuming E1 has already occurred, is given by
`P(E_2|E_1)=(P(E_2\ "and"\ E_1))/(P(E_1))`
Example 1
Let A denote the event 'student is female' and let B denote the event 'student is French'. In a class of `100` students suppose `60` are French, and suppose that `10` of the French students are females. Find the probability that if I pick a French student, it will be a girl, that is, find P(A|B).
Answer
Since `10` out of `100` students are both French and female, then
`P(A\ "and"\ B) = 10/100`
Also, `60` out of the `100` students are French, so
`P(B) = 60/100`
So the required probability is:
`P(A|B)=frac{P(A\ "and"\ B)}{P(B)}=frac{10/100}{60/100}=1/6`
Example 2
What is the probability that the total of two dice will be greater than `8`, given that the first die is a `6`?
Answer
Let E1 = first die is `6`;
Let E2 = total of two dice is ` > 8`
Then "E1 and E2" will be given by `(6, 3),\ (6, 4),\ (6, 5),\ (6, 6)`.
There are `36` possible outcomes when we throw 2 dice.
So
`P(E_2\ "and"\ E_1)=4/36=1/9`
Therefore
`P(E_2|E_1)` `= frac{P(E_2\ "and"\ E_1)}{P(E_1)}` `=frac{1/9}{1/6}=6/9` `=2/3`