Negative binomial distribution is a probability distribution of number of occurences of successes and failures in a sequence of independent trails before a specific number of success occurs. Following are the key points to be noted about a negative binomial experiment.

·        The experiment should be of x repeated trials.

·        Each trail have two possible outcome, one for success, another for failure.

·        Probability of success is same on every trial.

·        Output of one trial is independent of output of another trail.

·        Experiment should be carried out until r successes are observed, where r is mentioned beforehand.

Negative binomial distribution probability can be computed using following:

Formula

f(x;r,P)=x−1Cr−1×Pr×(1−P)x−rf(x;r,P)=x−1Cr−1×Pr×(1−P)x−r

Where −

·        xx = Total number of trials.

·        rr = Number of occurences of success.

·        PP = Probability of success on each occurence.

·        1−P1−P = Probability of failure on each occurence.

·        f(x;r,P)f(x;r,P) = Negative binomial probability, the probability that an x-trial negative binomial experiment results in the rth success on the xth trial, when the probability of success on each trial is P.

·        nCrnCr = Combination of n items taken r at a time.

Example

Robert is a football player. His success rate of goal hitting is 70%. What is the probability that Robert hits his third goal on his fifth attempt?

Solution:

Here probability of success, P is 0.70. Number of trials, x is 5 and number of successes, r is 3. Using negative binomial distribution formula, let’s compute the probability of hitting third goal in fifth attempt.

f(x;r,P)=x−1Cr−1×Pr×(1−P)x−r⟹f(5;3,0.7)=4C2×0.73×0.32=6×0.343×0.09=0.18522f(x;r,P)=x−1Cr−1×Pr×(1−P)x−r⟹f(5;3,0.7)=4C2×0.73×0.32=6×0.343×0.09=0.18522

Thus probability of hitting third goal in fifth attempt is 0.185220.18522.