Geometric Probability Calculator

Visualize and calculate the probability of the first success in a series of Bernoulli trials.

Input Parameters

%

Enter a value between 0 and 1.

Enter a positive integer.

Geometric Probability Result:

Probability Visualization:

Visual representation of geometric probability for the first success on trial .

Trials are represented along the x-axis. The highlighted bar indicates the geometric probability on the specified trial.

Understanding Geometric Probability

Geometric probability deals with the number of trials needed for the first success in a sequence of independent Bernoulli trials. Each trial has only two possible outcomes: success or failure, with the probability of success (p) remaining constant across all trials.

Formula

The probability that the first success occurs on the n-th trial is given by the formula:

$$P(X = n) = (1-p)^{n-1} \cdot p$$
  • P(X = n) is the geometric probability of the first success on the n-th trial.
  • p is the probability of success on a single trial.
  • n is the trial number on which the first success occurs.

Example

Suppose you are rolling a fair six-sided die until you roll a 6 for the first time. The probability of success (rolling a 6) on a single trial is p = 1/6. To find the probability that the first 6 occurs on the 3rd roll (n = 3), we use the formula:

$$P(X = 3) = (1 - \frac{1}{6})^{3-1} \cdot \frac{1}{6} = (\frac{5}{6})^{2} \cdot \frac{1}{6} \approx 0.1157$$

This means there is approximately an 11.57% chance that you will roll the first 6 on the third try.

Uses

Geometric probability is useful in scenarios where you are interested in the number of trials needed to achieve the first success, such as in quality control, sales, or games of chance. It helps in understanding the likelihood of achieving a desired outcome after a specific number of attempts.