Direct evaluation of cumulative probabilities

There is no simple formula for the negative binomial distribution's cumulative distribution function, \(F(x)\) but individual negative binomial probabilities can be added,

\[ F(x) = \sum_{u=k}^{\lfloor x \rfloor} {{u-1} \choose {k-1}} \pi^k(1-\pi)^{u-k} \]

where \(\lfloor x \rfloor \) denotes the smallest integer less than or equal to \(x\).

Cumulative probabilities from binomial distribution

If \(k\) is much smaller than \(x\), there is a simpler method since taking more than \(x\) trials to get the \(k\)'th success is equivalent to there being fewer than \(k\) successes in the first \(x\) trials.

\[ \begin{align} P(X \gt x) & = P(\text{fewer than } k \text{ successes in first } x \text{ trials}) \\ & = \sum_{v=0}^{k-1} {x \choose v} \pi^v (1-\pi)^{x-v} \end{align}\]

The cumulative probability is \(F(x) = 1 - P(X \gt x)\).

Cumulative probabilities in Excel

These formulae can be avoided if Excel is used. Again the function "NEGBINOM.DIST()" is used, but with its last parameter set to true. 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,

\(\mathbf{P( \text{no more than 16 trials} ) = P(X ≤ 16)}\), or
\(\mathbf{P( \text{no more than 11 failures} ) = P(X^* ≤ 11)}\)
Type in an Excel spreadsheet cell "=NEGBINOM.DIST(11, 5, 0.3, true)"

Question

If a fair six-sided die is rolled repeatedly, what is the probability that it will take more than 20 rolls before three sixes are observed?

(Solved in full version)