Geometric Distribution Probability Calculator

The geometric distribution is a discrete probability distribution that models the number of failures before the first success in a series of independent Bernoulli trials. This calculator computes geometric probabilities and visualizes the distribution.

Probability of success for a trial (p):

Number of failures until the first success (x):

P(X = ):

P(X < ):

P(X ≤ ):

P(X > ):

P(X ≥ ):

Understanding the Geometric Distribution

The Geometric distribution is a discrete probability distribution that models the number of failures before the first success in a sequence of independent trials, where each trial has the same probability of success. It is often used in scenarios like customer acquisition (how many people you need to contact before making a sale) or quality control (how many tests until you find a defective item).

Key Components of the Geometric Distribution

Probability of Success (p): The probability of success in each trial. For example, if the probability of making a sale in each customer call is 20%, then \( p = 0.20 \).
Number of Failures (X): The random variable \( X \) represents the number of failures before the first success. For example, if we want to know how many calls fail before making the first sale, \( X \) represents that number.

Geometric Distribution Formula

The probability mass function (PMF) for the Geometric distribution is given by the formula:

\[ P(X = x) = (1 - p)^x p, \quad \text{where} \ x = 0, 1, 2, \dots \]

The geometric formula calculates the probability of having exactly \( x \) failures before the first success, given a success probability \( p \).

Conditions for Using the Geometric Distribution

The Geometric distribution applies under specific conditions:

Independence: The trials must be independent, meaning the outcome of one trial does not affect the outcome of another.
Same Probability of Success: The probability of success \( p \) must remain constant for each trial.
Discreteness: The number of failures \( X \) must be a non-negative integer.

Step-by-Step Example: Finding Geometric Probability

Suppose we want to calculate the probability of observing exactly 3 failures before the first success in a process where the probability of success is 0.4.

Step 1: Identify the Key Parameters

In this case:

\( p = 0.4 \): The probability of success in each trial.
\( X = 3 \): The number of failures before the first success.

Step 2: Apply the Geometric Formula

Using the Geometric formula:

\[ P(X = 3) = (1 - 0.4)^3 \times 0.4 = 0.216 \times 0.4 = 0.0864 \]

Therefore, the probability of having exactly 3 failures before the first success is approximately 0.0864, or 8.64%.

Other Useful Probability Calculations

The Geometric distribution can also be used to calculate cumulative probabilities:

Less than \( x \) (P(X < \( x \))): The cumulative probability that the number of failures is less than a given value.
Greater than \( x \) (P(X > \( x \))): The cumulative probability that the number of failures is greater than a given value.
Less than or equal to \( x \) (P(X ≤ \( x \))): The cumulative probability that the number of failures is less than or equal to a given value.
Greater than or equal to \( x \) (P(X ≥ \( x \))): The cumulative probability that the number of failures is greater than or equal to a given value.

Practical Applications of the Geometric Distribution

The Geometric distribution is widely used in real-world applications, such as:

Sales and Marketing: Estimating the number of cold calls needed before making a sale.
Quality Control: Predicting the number of tests needed to find a defective product.
Recruitment: Calculating how many candidates you need to interview before finding a suitable hire.