Calculadora de Distribuição Geométrica
- Created by
- Renato Passos, Eng. de Software
- Reviewed by
- Renato Passos, Eng. de Software
Last updated: Apr 18, 2026
Formula
P(X=k) = (1−p)^(k−1)·p
About this calculator
The Geometric Distribution Calculator computes the probability that the first success occurs on a specific trial k, in a sequence of independent Bernoulli trials, each with success probability p. The formula used is P(X = k) = (1-p)^(k-1) * p, where k is the trial number (starting at 1) and p is between 0 and 1. This tool is useful for modeling situations where you want to know the chance of an event occurring for the first time on a given occasion.
The operation is simple: the user inputs the success probability p (as a decimal or percentage) and the trial number k. The calculator then applies the geometric distribution formula and returns the exact probability. Additionally, it displays the cumulative probability up to k, i.e., the probability that the first success occurs by trial k. This helps understand not only the point probability but also the cumulative chance.
Use this calculator in situations such as: estimating the probability of hitting a target on the 3rd shot, the chance of a machine failing for the first time on the 10th operation, or the probability of a customer making the first purchase on the 5th contact. It is common in quality control, reliability analysis, and time-to-event studies. The geometric distribution is discrete and models the number of trials until the first success.
Important precautions: the calculator assumes trials are independent and probability p is constant. If p is very small, the probability for large k can be very low. Also, note that the geometric distribution is memoryless: the probability of success on the next trial does not depend on previous ones. For events with replacement or without independence, other models are more appropriate.
Frequently asked questions
What does the value of k mean in the geometric distribution?
k represents the trial number on which the first success occurs. For example, k=3 means the first two trials were failures and the third was a success.
Can I use percentages for p?
Yes, you can enter p as a percentage (e.g., 20) or decimal (0.2). The calculator converts automatically.
Does the calculator provide the cumulative probability?
Yes, besides the exact probability P(X=k), it displays the cumulative probability P(X ≤ k), i.e., the chance that the first success occurs by trial k.
What is the difference between geometric and binomial distributions?
The geometric counts the number of trials until the first success, while the binomial counts the number of successes in a fixed number of trials.
What does 'memoryless' mean in the geometric distribution?
It means the probability of success on the next trial is always p, regardless of how many failures occurred before. For example, after 10 failures, the chance of success on the 11th is still p.