Geometric Probability Calculator

Calculate the probability of achieving the first success on a specific trial in a series of independent Bernoulli trials.

Geometric Probability Inputs

Enter the probability of success on a single trial (p) and the desired number of trials until the first success (k).

Probability of success (p):

Number of trials until first success (k):

Result

Probability of first success on trial :

Formula:

Probability Visualization

Understanding Geometric Probability

Geometric probability deals with the number of trials needed to get the first success in a sequence of independent Bernoulli trials. A Bernoulli trial is an experiment with only two outcomes: success or failure.

Formula

The probability mass function for geometric distribution is given by: $$P(X=k) = (1-p)^{k-1} \times p$$

P(X=k) is the probability that the first success occurs on the k-th trial.
p is the probability of success on each trial.
k is the number of trials until the first success (k = 1, 2, 3, ...).

Example

Suppose you are rolling a fair die until you get a 6. The probability of success (rolling a 6) is 1/6. What is the probability that the first 6 appears on the 3rd roll?

Probability of success (p) = 1/6
Number of trials until first success (k) = 3
Using the formula: P(X=3) = (1 - 1/6)^(3-1) * (1/6) = (5/6)^2 * (1/6) ≈ 0.1157

This calculator helps you quickly compute this probability for different values of p and k.