Conditional Probability:

Remember that the probability of an event occurring given that the other event has already occurred is termed as a conditional probability.

The probability which event B takes place, given that event A has already occurred is:

P(B|A) = P(A and B)/P(A)

The above formula is derived from the general multiplication principle and a little bit of algebra.

As we are given that the event A has occurred, we have a decreased sample space. Rather than the whole sample space S, we now contain a sample space of A as we know A has occurred. Therefore the old rule about being the number in the event divided by the number in sample space still applies. This is the number in A and B (should be in A as A has occurred) divided by the number in A. When divided numerator and denominator of the right hand side by the number in sample space S, then we have the probability of A and B divided by the probability of A.

Example: The question, ‘Do you smoke?’ was asked of 100 people. The outcomes are shown in the table below.

                Yes    No     Total
Male         19     41      60
Female     12     28      40
Total         31     69     100

1) Find out the probability of a randomly chosen individual being a male who smokes?

This is simply a joint probability. The number of Male and Smoke divided by the total = 19/100 = 0.19

2) Find out the probability of a randomly chosen individual being a male?

This is a total for male divided by the total = 60/100 = 0.60. As no mention is made up of smoking or not smoking, it comprises all the cases.

3) Find the probability of a randomly chosen individual smoking?

Again, as no mention is made of gender, this is the marginal probability, the total who smoke divided by the total = 31/100 = 0.31.

4) Find out the probability of a randomly chosen male smoking?

This time, you are told that you encompass a male - think of stratified sampling. Determine the probability that the male smokes? Well, 19 males smoke out of 60 males, therefore 19/60 = 0.31666...

5) Find out the probability that a randomly chosen smoker is male?

This time, you are told that you encompass a smoker and asked to determine the probability that the smoker is as well male. There are 19 male smokers out of 31 total smokers, therefore 19/31 = 0.6129

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