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×qx−1P(X=x)=p×qx−1

Where −

·        pp = probability of success for single trial.

·        qq = probability of failure for a single trial (1-p)

·        xx = the number of failures before a success.

·        P(X−x)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×qx−1P(X=x)=p×qx−1

Where −

·        p=30%=0.3p=30%=0.3

·        x=5x=5 = the number of failures before a success.

Therefore, the required probability:

P(X=5)=0.3×(1−0.3)5−1,=0.3×(0.7)4,≈0.072≈7.2%