# Statistics - Geometric Distribution

Geometric distribution is a special case of negative binomial distribution with 1 success out of specified,n trials. In other words, it is a probability distribution of number of occurences of successes and failures in a sequence of independent trails before a first success occurs. Following are the key points to be noted about a geometric experiment.

• The experiment should be of x repeated trials.

• Each trail have two possible outcome, one for success, another for failure.

• Probability of success is same on every trial.

• Output of one trial is independent of output of another trail.

• Experiment should be carried out until first success is observed.

Geometric distribution probability can be computed using following:

## Formula

${ g(x; P) = P \times (1-P)^{x-1} }$

Where −

• ${x}$ = Total number of trials.

• ${P}$ = Probability of success on each occurence.

• ${1-P}$ = Probability of failure on each occurence.

• ${g(x; P)}$ = Geometric distribution probability, the probability that an x-trial negative binomial experiment results in the first success on the xth trial, when the probability of success on each trial is P.

## Example

Robert is a football player. His success rate of goal hitting is 70%. What is the probability that Robert hits his first goal on his fifth attempt?

Solution:

Here probability of success, P is 0.70. Number of trials, x is 5. Using geometric distribution formula, let's compute the probability of hitting first goal in fifth attempt.

${ g(x; P) = P \times (1-P)^{x-1} \\[7pt] \implies g(x; P) = 0.7 \times 0.3^4 \\[7pt] \, = 0.7 \times 0.0081 \\[7pt] \, = 0.00567 }$

Thus probability of hitting first goal in fifth attempt is ${ 0.00567 }$.