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Geometric Probability Distribution
Statistics - Geometric Probabipty Distribution
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
${P(X=x) = p imes q^{x-1} }$
Where −
${p}$ = probabipty of success for single trial.
${q}$ = probabipty of failure for a single trial (1-p)
${x}$ = the number of failures before a success.
${P(X-x)}$ = Probabipty 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 probabipty 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 probabipty experiment of getting the first success in 5 trials. The problem statement also suggests the probabipty distribution to be geometric. The probabipty of success is given by the geometric distribution formula:
${P(X=x) = p imes q^{x-1} }$
Where −
${p = 30 \% = 0.3 }$
${x = 5}$ = the number of failures before a success.
Therefore, the required probabipty:
$ {P(X=5) = 0.3 imes (1-0.3)^{5-1} , \[7pt] , = 0.3 imes (0.7)^4, \[7pt] , approx 0.072 \[7pt] , approx 7.2 \% }$ Advertisements