Excel

Excel has a function that will evaluate either single negative binomial probabilities or cumulative probabilities, "NEGBINOM.DIST()". For example, if the probability of success is π = 0.3 and we are interested in how long it takes to get \(k = 5\) successes,

\(\mathbf{P( \text{exactly 12 trials} ) = P(X = 12)}\), or
\(\mathbf{P( \text{exactly 7 failures} ) = P(X^* = 7)}\)
Type in an Excel spreadsheet cell "=NEGBINOM.DIST(7, 5, 0.3, false)"

Excel is implicitly using the second form of binomial distribution for the number of failures before the \(k\)'th success.

On a scientific calculator

Negative binomial probabilities can also be found on a scientific calculator directly using the formula for its probability function.

Question

If a coin is repeatedly tossed, what is the probability that the second head will appear on the fifth toss of the coin?

(Solved in full version)