Geometric Distribution Calculator

The geometric distribution models the number of independent Bernoulli trials needed to get the first success, where each trial has a constant success probability denoted as p. The primary mathematical formulas governing this calculator are: Geometric Probability is calculated as P(X = k) = ((1 - p)^k) × p, where k represents the number of failures before the first success. The Expectation (Mean), representing the average number of failures expected before a success, is given by E(X) = (1 - p) / p. The Variance is calculated as Var(X) = (1 - p) / p², and the Standard Deviation is simply the square root of the variance.

Imagine you are launching a targeted online marketing campaign where the conversion rate (success probability, p) for a specific user action is 20%, or 0.20. You want to know the exact probability that your ad will experience precisely 3 failures before securing its very first successful conversion. First, identify your inputs: success probability p = 0.20 and failures k = 3. Next, apply the probability formula: P(X = 3) = ((1 - 0.20)³) × 0.20. This simplifies to (0.80³) × 0.20, which equals 0.512 × 0.20 = 0.1024, or a 10.24% chance. To find the expected number of failures before the first success, compute E(X) = (1 - 0.20) / 0.20 = 0.80 / 0.20 = 4 failures.

When working with geometric distributions, ensure that your trials meet the core criteria: each trial must be completely independent of the others, and the probability of success must remain strictly constant across every single trial. Additionally, double-check whether your specific problem asks for the number of failures before the first success or the total number of trials needed to achieve that first success, as shifting your definitions by one integer will alter your results.