AP Stats Geometric Distribution Calculator

How many trials until the first success? Enter the success probability and the trial number, and this AP Stats Geometric Distribution Calculator gives the exact probability, the cumulative probability, and the expected number of trials.

What Is the Geometric Distribution?

The geometric distribution gives the probability that the first success happens on a given trial, when trials are independent and each has the same probability of success. If you roll a die until you get a 6, the chance the first 6 comes on the third roll is (5/6) squared times (1/6). The expected number of trials until the first success is 1/p.

How to Use This Calculator

1. Enter p, the probability of success on a single trial, as a decimal. 2. Enter x, the trial on which the first success occurs. 3. Press Calculate. 4. Read P(X = x) for exactly that trial, P(X ≤ x) for success on or before that trial, and the mean number of trials until the first success.

How to Calculate a Geometric Probability

Formulas: P(X = x) = (1 − p)^x−1 × p, mean = 1 / p, and cumulative P(X ≤ x) = 1 − (1 − p)^x. The trials must be independent with the same p on every trial.

Worked example: p = 0.25, x = 4. P(X = 4) = (0.75)³ × 0.25 = 0.1055. P(X ≤ 4) = 1 − (0.75)⁴ = 0.6836. Mean = 1 / 0.25 = 4 trials. Try these values in the calculator to see the same result and the probability bars.

Tips and Common Mistakes

The exponent is x − 1, not x, because the first x − 1 trials are failures and only the last trial is a success. Do not confuse geometric with binomial: geometric counts trials until the first success with no fixed number of trials, while binomial counts successes in a fixed number of trials. For "at least" questions, use the complement: P(X > x) = (1 − p)^x.

Frequently Asked Questions

What does a geometric random variable count?

It counts the number of trials until the first success, including the successful trial itself.

What is the difference between geometpdf and geometcdf?

geometpdf gives P(X = x), the probability the first success is exactly on trial x. geometcdf gives P(X ≤ x), the probability the first success happens on or before trial x.

Why is the mean 1/p?

If success is likely, you expect to wait few trials; if it is rare, you expect to wait many. With p = 0.25, the average wait is 4 trials.

Can x be 0?

No. The first success cannot happen on trial 0. The smallest possible x is 1, which means success on the very first trial, with probability p.

When is a geometric model not appropriate?

When trials are not independent, when p changes from trial to trial, or when there is a fixed number of trials and you are counting successes instead of waiting for the first one.

What is the TI-84 equivalent?

On a TI-84, use 2nd, DISTR, geometpdf(p, x) for the exact probability and geometcdf(p, x) for the cumulative probability.

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