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Statistics & Data Science · pattern A, Numeric fields

How many trials until the first success.

Enter the probability of success per trial and the trial number for the probability that the first success falls exactly there, with the expected wait and the cumulative probability.

Inputs

The formula used

P(X = k) = (1 − p)^(k−1) p

Counts trials up to and including the first success. A variant counts failures before the first success, which shifts every result by one — check which convention your module uses.

P(X = k)

0.102400

Expected number of trials, 1/p
5.000
Variance, (1−p)/p²
20.000
Standard deviation
4.472
P(X ≤ k)
0.590400
P(X > k)
0.409600
Trials for a 95% chance of success
14

The distribution has no memory.

After twenty failures, the probability that the next trial succeeds is still p. Past failures change nothing, which is counter-intuitive and is exactly the gambler's fallacy stated formally. The expected wait of 1/p is also frequently misread: with p = 0.2 the average wait is five trials, but the single most likely outcome is success on the very first one.

Questions about geometric distribution

What does memoryless mean?
That the probability of success on the next trial is independent of how many failures preceded it. The distribution forgets its history entirely.
Why is the most likely value 1 when the mean is 1/p?
Because the distribution is heavily skewed: the single most probable outcome is an immediate success, while a long tail of late successes pulls the mean upward.
What is the negative binomial?
The generalisation to the wait until the rth success. The geometric distribution is the special case with r = 1.

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Statistical output here is for coursework and learning. Reported p-values and intervals assume the conditions stated on the page; nothing here checks whether those conditions hold for your data.