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:

Formula

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

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.

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.

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

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

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