On a computer
Excel has a function that will evaluate either single negative binomial probabilities or cumulative probabilities, "NEGBINOM.DIST()". To find the probability that the \(k\)'th success in a series of Bernoulli trials occurs on the \(x\)'th trial, \(p(x)\), the function takes four parameters:
(Excel is therefore implicitly using the second form of binomial distribution for the number of failures before the \(k\)'th success.)
By hand
Of course, negative binomial probabilities can also be found on a scientific calculator directly using the formula for its probability function.
Example
If a coin is repeatedly tossed, what is the probability that the second head will appear on the fifth toss of the coin?
If \(X\) denotes the toss when the second head appears,
\[ X \;\; \sim \; \; \NegBinDistn(2, 0.5) \]The answer is therefore
\[ p(5) = {{5-1} \choose {2-1}} {0.5}^2(1-{0.5})^{5-2} = {4 \choose 1} {0.5}^5 = 0.125 \]In Excel, this could be found by typing in a spreadsheet cell
=NEGBINOM.DIST(3, 2, 0.5, false)