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The geometric distribution is a special case of the negative binomial distribution. It deals with the number of trials required for a single success. Thus, the geometric distribution is a negative binomial distribution where the number of successes (r) is equal to 1.
Formula
${P(X=x) = p times q^{x-1} }$
Where −
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${p}$ = probability of success for single trial.
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${q}$ = probability of failure for a single trial (1-p)
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${x}$ = the number of failures before a success.
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${P(X-x)}$ = Probability of x successes in n trials.
Example
Problem Statement:
In an amusement fair, a competitor is entitled for a prize if he throws a ring on a peg from a certain distance. It is observed that only 30% of the competitors are able to do this. If someone is given 5 chances, what is the probability of his winning the prize when he has already missed 4 chances?
Solution:
If someone has already missed four chances and has to win in the fifth chance, then it is a probability experiment of getting the first success in 5 trials. The problem statement also suggests the probability distribution to be geometric. The probability of success is given by the geometric distribution formula:
${P(X=x) = p times q^{x-1} }$
Where −
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${p = 30 % = 0.3 }$
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${x = 5}$ = the number of failures before a success.
Therefore, the required probability:
, = 0.3 times (0.7)^4, \[7pt]
, approx 0.072 \[7pt]
, approx 7.2 % }$
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