Example: We are interested in being able to compute the probability of x successes in n Bernoulli trails. For example, suppose that in a certain population 52 percent of all recorded births are males. We interpret this to mean that the probability of a recorded male birth 0.52. If we randomly select 5 birth records from this population, what is the probability that exactly 3 of the records will be for male births?
Solution: Let us designate the occurrence of a record for a male as a “success”, and hasten to add that this is an arbitrary designation for purposes of clarity and convenience and does not reflect an opinion regarding the relative merits of male versus female births. The occurrence of a birth record for a male will be designated a success, since we are looking for birth records for males. If we are looking for the birth records of females, these would be designated as successes, and birth records of males would be designated as failures.
It will also be convenient to assign the number 1 to a success (record of a male birth) and the number 0 to a failure (record of a female birth).
The process that eventually results in a birth record, we consider to be a Bernoulli process.
Suppose, the 5 records selected resulted in this sequence of sexes
MFMMF
In coded form, we would write this as
10110
Since the probability of a success is denoted by p and the probability of a failure is denoted by q, the probability of the above sequence of outcomes is found by means of the multiplication rule to be
P(1,0,1,1,0) = pqppq = q2 p3
The multiplication rule is appropriate for computing this probability since we are seeking the probability of a male, and a female, and a male, and a male, and a female, in that order or, in other words, the joint probabilities of the 5 events. So the probability of 3 records as male births is (0.52)3 (1-0.52)2 = (0.52)3 . (0.48)2.
