Discussion
What is the probability of observing exactly k successes in n trials in a binomial distribution with parameter p ?
Answer: D
The probability is \({}^{n}C_{k}\, p^{k}(1-p)^{n-k}\).
Any particular sequence of k successes and n - k failures has probability \(p^{k}(1-p)^{n-k}\), and there are \({}^{n}C_{k}\) ways of choosing which k of the n trials are the successes.
Without the \({}^{n}C_{k}\) factor you get the probability of one particular sequence only. \(p(1-p)^{k}\) and \({}^{n}C_{k}\, p^{n}(1-p)^{k}\) put the exponents in the wrong places.
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