Geometric Distribution Calculator

What Is the Geometric Distribution?

The geometric distribution models the number of independent Bernoulli trials needed until the first success, or equivalently the number of failures before the first success. Classic examples include rolling a die until you get a six, flipping a coin until heads, or testing products until the first defect is found. For related discrete models, see the Binomial Probability Distribution Calculator or Negative Binomial Distribution Calculator.

Key Formulas

With $x$ failures before the first success and success probability $p$:

$$P(X = x) = (1-p)^x \cdot p$$

$$\mu = \frac{1-p}{p}, \quad \sigma^2 = \frac{1-p}{p^2}, \quad \sigma = \sqrt{\frac{1-p}{p^2}}$$

Example: Rolling a Die

Rolling a fair six-sided die until the first six has $p = 1/6$. The probability of exactly one failure before the first success ($x = 1$) is $P(X = 1) = (5/6) \times (1/6) \approx 0.1389$, or about 13.89%. The mean number of failures is $(1-p)/p = 5$.

Related Tools

• Exponential Distribution Calculator
• Expected Value Calculator
• Binomial Probability Distribution Calculator