Advanced Geometric Distribution Calculator

Calculator Inputs

Variable interpretation

Calculation mode

Success probability (p)

x value

Lower bound (a)

Upper bound (b)

Target cumulative probability

Generated table upper limit

Example Data Table

Sample values below use p = 0.30 with X as the trial number of the first success.

x	P(X = x)	P(X ≤ x)
1	0.3	0.3
2	0.21	0.51
3	0.147	0.657
4	0.1029	0.7599
5	0.07203	0.83193
6	0.050421	0.882351

Formula Used

1) Trial number of the first success

PMF: P(X = x) = (1 - p)^{x - 1} × p, for x ≥ 1

CDF: P(X ≤ x) = 1 - (1 - p)^x

Survival: P(X > x) = (1 - p)^x

2) Failures before the first success

PMF: P(X = x) = (1 - p)^x × p, for x ≥ 0

CDF: P(X ≤ x) = 1 - (1 - p)^{x + 1}

Survival: P(X > x) = (1 - p)^{x + 1}

3) Distribution measures

Mean: 1/p for trial-count form, or (1 - p)/p for failures-count form

Variance: (1 - p)/p²

Interval probability: P(a ≤ X ≤ b) = F(b) - F(a - 1)

How to Use This Calculator

Choose whether X represents the trial number of the first success or the number of failures before the first success.
Enter the success probability p. This must stay between 0 and 1.
Select a calculation mode such as PMF, CDF, survival, interval probability, percentile, or summary statistics.
Fill in the needed value fields. For example, PMF uses x, interval mode uses a and b, and percentile mode uses the target cumulative probability.
Set the generated table upper limit to create a supporting probability table for review and export.
Press Calculate Distribution to display the result above the form, then review the measures and the generated table.
Use the CSV and PDF buttons after calculation to save the result summary and the generated distribution table.

Frequently Asked Questions

1) What does the geometric distribution measure?

It measures how long you wait for the first success in repeated independent Bernoulli trials, or how many failures occur before that first success.

2) When should I use the trial-number version?

Use it when the first success can happen on trial 1, 2, 3, and so on. This form starts counting at one.

3) When should I use the failures-before-success version?

Use it when zero is a valid outcome. It counts unsuccessful trials before the first success appears.

4) What assumptions must hold for the model?

Trials must be independent, each outcome must be success or failure, and the success probability must remain constant across all trials.

5) What is the survival probability?

It gives the chance that the first success occurs after the selected value, which is useful for waiting-time and reliability questions.

6) Why do the two interpretations have different means?

The trial-number form includes the success trial itself, while the failures form does not. That shifts the mean by exactly one.

7) How is the percentile calculated?

The calculator returns the smallest integer whose cumulative probability reaches or exceeds the target level you entered.

8) Why may the generated table not sum exactly to one?

A limited table can omit remaining tail probability, and rounded display values can create very small visual differences.