Statistics - Negative Binomial Distribution

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:


${ f(x; r, P) = ^{x-1}C_{r-1} \times P^r \times (1-P)^{x-r} }$

Where −

  • ${x}$ = Total number of trials.

  • ${r}$ = Number of occurences of success.

  • ${P}$ = Probability of success on each occurence.

  • ${1-P}$ = Probability of failure on each occurence.

  • ${f(x; r, P)}$ = 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.

  • ${^{n}C_{r}}$ = Combination of n items taken r at a time.


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?


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.

${ f(x; r, P) = ^{x-1}C_{r-1} \times P^r \times (1-P)^{x-r} \\[7pt] \implies f(5; 3, 0.7) = ^4C_2 \times 0.7^3 \times 0.3^2 \\[7pt] \, = 6 \times 0.343 \times 0.09 \\[7pt] \, = 0.18522 }$

Thus probability of hitting third goal in fifth attempt is $ { 0.18522 }$.