Geometric Distribution Formula
The geometric distribution denotes the number of failures before we get a success in a series of Bernoulli trials. It is a special case of the negative binomial distribution. In the geometric distribution formula, we go for a number of trials required for a single success. The geometric distribution formula is a discrete random variable having a probability density function of the following form that is said to have a geometric distribution.
What Is Geometric Distribution Formula?
The formula of geometric distribution is given below:
P(X = x) = q^{(x1)}p
Where,
p = probability of success for single trial
q = probability of failure for single trial (q = 1p)
x = number of failures before success
P(X = x) = Probability of x successes in n trials.
Two other formulas used in geometric distribution:
Mean = \(\dfrac{1}{p}\)
Variance = \(\dfrac{1p}{p^2}\)
Solved Examples Using Geometric Distribution Formula

Example 1
In a school competition, a student is entitled to a prize if he/she throws a ring on a peg from a certain distance. It is observed that only 40% of the students can do this. If someone has already missed 4 chances from 5 chances, what is the probability of that student winning the prize?
Solution: This is a case of geometric distribution as in it we are checking on the number of trials for success. Someone has already missed 4 times, maybe he succeeds the 5th time.
To find: Probability, mean, and variance.
Given,
p = 0.4 , x = 5, q= 1 p= 0.6
Using geometric distribution formula,
P(X = x) = q^{(x1)}p
P (X=5) = 0.4 × 0.6 ^{(51)}
P (X=5) = 0.4 × 0.6 ^{(4)}
P (X=5) = 0.052 = 5.2%Answer: The probability of winning the prize is 5.2%

Example 2:
Calculate the mean and variance for Q.1
Solution:
To find: The mean and variance
Answer: The mean is 2.5 and the variance is 3.75.
Given,
p = 0.4 , q= 1 p= 0.6
Using formulas for mean and variance.
Mean = \(\dfrac{1}{p}\)
Variance = \(\dfrac{1p}{p^2}\)
Now,
Mean = \(\dfrac{1}{0.4}\)
Mean = 2.5
Now,
Variance = \(\dfrac{10.4}{0.4^2}\)
Variance = 3.75