Geometric Distribution CalculatorCalculator
Calculate exact probability P(X = k), cumulative P(X ≤ k), and expected trials to first success in repeated Bernoulli trials.
Bernoulli Trial Parameters
Trial Probability Results
Geometric Distribution Calculator - Trials to First Success: Compute trial probabilities until first success, cumulative likelihood P(X ≤ k), and expected trials 1/p. Free, private, and simple to use in your browser.
How to Use the Geometric Distribution Calculator
Define Probability of Success (p)
Enter the constant single-trial likelihood of success between 0 and 1.
Choose Target Trial (k)
Specify on which trial number you are evaluating the first occurrence of success.
Evaluate Exact Probability
Read P(X = k) representing k - 1 failures followed immediately by a success.
Inspect Cumulative Horizon
Check P(X ≤ k) to determine the probability of succeeding within k total attempts.
Statistical Modeling & Mathematical Analysis
The geometric distribution is a discrete probability distribution that models the number of Bernoulli trials required to achieve the very first success. Each trial is assumed to be strictly independent with a constant probability of success p and failure probability q = 1 - p.
Under the standard textbook formulation (number of trials until the first success, X ∈ {1, 2, 3, ...}), the probability mass function is given by: P(X = k) = (1 - p)^(k - 1) · p. This reflects the joint probability of experiencing k - 1 consecutive failures followed by a success on trial k.
The cumulative distribution function (CDF) has a particularly clean closed-form expression: P(X ≤ k) = 1 - (1 - p)^k. This directly answers the practical question: "What is the probability of succeeding within k total attempts?" The expected value (mean number of trials until first success) is simply E[X] = 1 / p.
Crucially, the geometric distribution is the ONLY discrete distribution that possesses the memoryless property: P(X > s + t | X > s) = P(X > t). If you have already failed s times, the probability that it takes an additional t trials to succeed is completely unaffected by your past failures.
Key Analytical Advantages
Bernoulli Horizon Modeling
Determines how many attempts are required to land an interview, sale, or lottery win.
Closed-Form Precision
Exact closed-form formulas eliminate rounding inaccuracies in large trial counts.
Memoryless Property Insights
Explains why previous consecutive failures do not increase future success odds.
Expected Value & Variance
Reports expected trials 1/p and standard deviation for resource allocation planning.
Fast Client-Side Engine
Computes probability curves across multiple candidate trial horizons instantly.
Confidential Sales Modeling
Proprietary customer conversion rates are processed securely inside your browser.
Frequently Asked Questions
The memoryless property states that past failures do not influence the probability of future success: P(X > s + t | X > s) = P(X > t). If you flip a coin and get tails 5 times, the probability of getting heads on the next flip remains exactly 0.5.
The Binomial distribution counts the number of successes in a fixed number of trials n. The Geometric distribution counts the number of trials needed to achieve the first single success.
The expected number of trials is E[X] = 1 / p = 1 / 0.25 = 4 trials.
P(X ≤ k) is the cumulative probability that your first success occurs on or before trial k. It equals 1 - (1 - p)^k.
In the standard formulation (trials until first success), k must be at least 1 (X ∈ {1, 2, 3, ...}). In an alternative formulation that counts only failures before success (Y = X - 1), Y can be 0.