An Explanation of Binomial Distribution Part II

Example

Let P (B_) = 3/4; P (bb) = 1/4; n = 4 which is the family size; r = o which is the number of Bb offspring or successful events.

The binomial distribution is:

This formula will tell us the probability of r ‘successful’ events out of a total of n events. We will define p as the probability of a ‘success’ and q as the probability of a ‘failure’. p + q = 1 and n = r + (n-r).

In our example we have a family size of 4 or 4 events, n = 4. We want to find the probability of r = o or zero successes and 4 failures. Four offspring of the bb genotype is the equivalent of four failures and the probability of each individual failure = 1/4. P (event) = P (families) family size = 4, = # of trials. We are expanding the binomial expression (p+q).
genotn! = 4 x 3 x 2 x 1 = 24
genotr! = 0! = 1
genot(n-r)! = (4-0)! = 4! = 24

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Our expectation is that the proportion of families of size 4 which contain four bb offspring is 1/256.