Geometric Distribution
Calculate geometric distribution probabilities P(X=x), mean, variance, and standard deviation. Enter probability of success (p) and number of failures (x) for instant results with bar charts.
About This Calculator
The Geometric Distribution Calculator helps you compute probabilities for the geometric distribution — a discrete probability distribution that models the number of failures before the first success in a sequence of independent Bernoulli trials. This calculator is ideal for students learning probability theory, statisticians analyzing trial-based processes, and professionals working with quality control, reliability testing, or risk assessment.
The geometric distribution follows the formula P(X = x) = (1-p)^x × p, where p is the probability of success on a single trial and x is the number of failures before the first success. The mean (expected value) is μ = (1-p)/p, the variance is σ² = (1-p)/p², and the standard deviation is σ = √((1-p)/p²). The distribution is memoryless, meaning past failures do not affect the probability of future success.
Geometric distribution is used globally in fields such as manufacturing quality control (number of items inspected until finding a defect), telecommunications (bits transmitted until an error), healthcare (patients tested until a positive diagnosis), and finance (trades until a profitable outcome). Its applications are universal and not tied to any specific country or region.
Example
If you flip a fair coin (p = 0.5 for heads), what is the probability of getting 2 tails before the first head? Using the formula P(X = 2) = (1-0.5)² × 0.5 = 0.25 × 0.5 = 0.125 = 12.5%. The mean number of tails before the first head is (1-0.5)/0.5 = 1, and the variance is (1-0.5)/0.5² = 2.
Frequently Asked Questions
What is geometric distribution?
Geometric distribution is a probability distribution that models the number of failures before the first success in a sequence of independent Bernoulli trials. Each trial has the same probability of success p. For example, it can model the number of tails before getting the first head when flipping a fair coin, or the number of defective items inspected before finding the first good one.
What is the geometric distribution formula?
The geometric distribution formula is P(X = x) = (1 - p)^x × p, where p is the probability of success on a single trial and x is the number of failures before the first success. The mean is (1-p)/p, the variance is (1-p)/p², and the standard deviation is the square root of the variance.
What is the difference between geometric and binomial distribution?
The geometric distribution models the number of trials needed to get the first success, while the binomial distribution models the number of successes in a fixed number of trials. In geometric distribution, the number of trials is variable and stops at the first success. In binomial distribution, the number of trials is fixed and you count how many succeed.
What is the memorylessness property of geometric distribution?
The geometric distribution is memoryless, meaning that the probability of needing additional trials beyond a certain point does not depend on how many failures have already occurred. Formally, P(X > m + n | X > m) = P(X > n). This property is shared with the exponential distribution.
What are real-world examples of geometric distribution?
Common examples include: rolling a die until getting a 6; a couple planning children until having a boy; testing items on an assembly line until finding a defective one; transmitting data bits until the first error; and interviewing voters until finding one who supports a specific candidate.
How do you calculate the mean and variance of geometric distribution?
The mean (expected value) of geometric distribution is μ = (1-p)/p. The variance is σ² = (1-p)/p². The standard deviation is the square root of the variance. For example, with p = 0.5, the mean is 1, meaning on average you expect 1 failure before the first success.
What does the number of failures (x) mean in geometric distribution?
In geometric distribution, x represents the number of failures that occur before the first success. For example, if x = 2, it means two failures happened before the first successful trial. The probability P(X = x) gives the likelihood of observing exactly x failures followed by a success.
Can the probability of success be zero or one in geometric distribution?
No, the probability of success p must be strictly greater than 0 and less than or equal to 1. If p = 0, success is impossible and the distribution is undefined. If p = 1, success is guaranteed on the first trial, so P(X = 0) = 1 and the mean is 0 with zero variance.