Example: : In the game of craps, you decide to play until you
lose 5 games. You wonder how many games you will play with this
termination rule. The probability of losing any one game is
0.5071. The games are a series of independent Bernoulli trials,
and the random variable is the number of wins until the fifth
loss. This is a situation described by the negative binomial
distribution.
For the example, we perversely defined success as a “loss”
with *p *the probability of a success equal to 0.5071
for the example. The random variable is the number of trials
that result in 0 before the *r*th 1 is observed. For
this case, *r* = 5.
It is important to remember that the random variable is not
the total number of trials but the number of failed trials before
the *r*th success. In the solution, the entry for 0 describes
the probability that the first five plays were losses and there
were no wins. |