Statistics - Geometric Probability 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

${P(X=x) = p \times q^{x-1} }$

Where −

${p}$ = probability of success for single trial.
${q}$ = probability of failure for a single trial (1-p)
${x}$ = the number of failures before a success.
${P(X-x)}$ = 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 −

${p = 30 \% = 0.3 }$
${x = 5}$ = the number of failures before a success.

Therefore, the required probability:

$ {P(X=5) = 0.3 \times (1-0.3)^{5-1} , \\[7pt] \, = 0.3 \times (0.7)^4, \\[7pt] \, \approx 0.072 \\[7pt] \, \approx 7.2 \% }$

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