Website owner: James Miller
Theorem. Let X is the number of successes in n independent Bernoulli experiments. Let the probability of success in a single experiment be p. Then the probability of exactly k successes in n experiments is given by
1) P(X = k) = nCkpkqn-k
where q = 1 - p and nCk = n! / [k!(n-k)!].
Derivation. We compute the probability Pk of precisely k occurrences of a certain event A in n independent tests, in each of which the probability p of the occurrence of this event is the same. We denote by A the event that event A does not occur. It is obvious that
P(A) = 1 - P(A) = 1 - p
From the definition of the independence of experiments it is easy to see that the probability of any specific sequence consisting of k occurrences of A and n-k nonoccurrences of A is equal to
2) pk(1- p)n-k
Thus, for example, for n = 5 and k = 2 the probability of getting the sequence AAAAA will be p(1 - p)p(1 - p)(1 - p) = p2(1 - p)3.
By the theorem on the addition of probabilities, Pk will be equal to the sum of the probabilities of all sequences with k occurrences and n - k nonoccurrences of the event A, i.e. Pk will be equal to the product of the number of such sequences by pk(1- p)n-k. The number of such sequences is equal to the number of combinations of n things taken k at a time since the k positive outcomes may occupy any k places in the sequence of n trials (i.e. the number of combinations corresponds to all combinations of the AAAAA ... AA sequences). This number of combinations is given by nCk = n! / [k!(n-k)!]. Thus
P(X = k) = nCkpkqn-k
Mostly excerpted. Source: Mathematics, Its Content, Methods and Meaning, Vol. 2
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