Monday, May 30, 2011

Section 2.4 - The Binomial Distribution—Bernoulli Trials

A Bernoulli trial produces an outcome which satisfies a criterion with probability \(p\), and so fails to satisfy the criterion with compliment probability \(q = 1 - p\). What is the probability that \(n\) successive, independent Bernoulli trials results in exactly \(k\) satisfactory outcomes?

The compound outcome is an arrangement of \(k\) satisfactory outcomes and \(n - k\) unsatisfactory outcomes. Because the Bernoulli-trial outcomes are independent, the outcome probabilities are multiplied to get the combined-outcome probability, and because multiplication is commutative, any individual compound outcome has probability \(p^kq^{n-k}\). There are \(C(n, k)\) such compound outcomes, and the probability that \(n\) successive, independent Bernoulli trials results in exactly \(0 \leq k \leq n\) satisfactory outcomes when satisfactory outcomes have probability \(p\) is \[ b(k; n, p) = C(n, k)p^kq^{n - k}\] The compound-outcome sample space is not uniform. Even if \(p = 1/2\),? \(C(n, k)\) varies with \(k\); the variation becomes more complex when \(p\neq q\).