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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.
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The experiment should be of x repeated trials.
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Each trail have two possible outcome, one for success, another for failure.
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Probability of success is same on every trial.
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Output of one trial is independent of output of another trail.
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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 −
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${x}$ = Total number of trials.
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${r}$ = Number of occurences of success.
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${P}$ = Probability of success on each occurence.
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${1-P}$ = Probability of failure on each occurence.
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${f(x; r, P)}$ = 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.
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${^{n}C_{r}}$ = 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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