Enter a probability of success (p) and a number of trials (x) to get the full geometric distribution breakdown. The calculator returns P(X = x), P(X ≤ x), P(X ≥ x), plus the mean, variance, and standard deviation — covering both the number-of-trials and number-of-failures conventions.
P(X = x)
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P(X ≤ x)
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P(X ≥ x)
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Mean (μ)
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Variance (σ²)
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Standard Deviation (σ)
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A geometric experiment is a statistical experiment with repeated, independent trials where each trial has exactly two outcomes — success or failure — with a constant probability of success on every trial. The experiment continues until the first success is observed. The random variable X records either the trial number of that first success or the number of failures before it, depending on convention.
The probability of success, p, is the fixed chance that any single trial results in a success. It must be greater than 0 and at most 1. For example, if you roll a fair die and define success as rolling a 6, then p = 1/6 ≈ 0.1667.
Under the trials convention, x is the total number of trials conducted up to and including the first success, so x ≥ 1. Under the failures convention, x is the count of failed attempts before the first success, so x ≥ 0. Choosing the correct convention determines which formula is applied.
Under the trials convention, P(X = x) = (1 − p)^(x−1) × p. Under the failures convention, P(X = x) = (1 − p)^x × p. The cumulative probability P(X ≤ x) sums all individual probabilities from 1 (or 0) up to x, and P(X ≥ x) = 1 − P(X ≤ x − 1).
Under the trials convention, the mean μ = 1/p. Under the failures convention, μ = (1 − p)/p. For example, if p = 0.25 the expected number of trials until the first success is 1/0.25 = 4.
The variance is σ² = (1 − p)/p² for both conventions (the shift between them does not change the spread). The standard deviation σ = √((1 − p)/p²). A lower probability of success p gives a higher variance, reflecting greater uncertainty about when the first success will occur.
One convention defines X as the total number of trials until the first success (X ≥ 1), used widely in North American textbooks. The other defines X as the number of failures before the first success (X ≥ 0), common in European literature and software like R's dgeom(). The formulas differ by a shift of 1 in the exponent, so always check which convention your course or software uses.
Yes — the geometric distribution is the only discrete probability distribution with the memoryless property. This means that given you have already had k failures, the probability of needing m more trials is the same as if you were starting fresh. Formally, P(X > k + m | X > k) = P(X > m).