Geometric Distribution Calculator

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Trials convention: X counts each attempt including the success. Failures convention: X counts only the failed attempts before the first success. The two are related by X_failures = X_trials - 1.
Probability of success on each independent Bernoulli trial. Must be between 0 and 1 (exclusive).
The specific value to evaluate. For the trials convention k must be 1 or more; for the failures convention k can be 0 or more.
trials
P(X = k)Moderate probability
0.1029

Probability that the first success occurs exactly at trial k

P(X ≤ k)0.7599
P(X < k)0.657
P(X ≥ k)0.343
P(X > k)0.2401
Mean (expected value)3.3333
Variance7.7778
Standard deviation2.7889
P(X = k)0.1029
P(X ≤ k)0.7599

P(X = 4) = 10.29% with p = 0.3

  • There is a 10.29% chance the first success happens exactly at trial 4.
  • By trial 4, the cumulative probability of at least one success is 75.99%.
  • With p = 0.3, you can expect the first success after about 3.33 trials on average.

Next stepTo find how many trials you need to be 90% sure of at least one success, solve 1 - (1-p)^n >= 0.9, which gives n >= 7.

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