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:

\, = 0.3 \times (0.7)^4, \\[7pt]

\, \approx 0.072 \\[7pt]

\, \approx 7.2 \% }$

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