What the Geometric Distribution Models
The geometric distribution models how long it takes to get the first success in a sequence of independent trials. Each trial must have the same probability of success, p.
This formula uses the convention where X is the trial number of the first success. For example, X = 3 means two failures followed by one success.
Two Common Conventions
| Convention | Support | Exact formula |
|---|---|---|
| Trials until first success | k = 1, 2, 3, ... | P(X = k) = (1 − p)k − 1p |
| Failures before first success | y = 0, 1, 2, ... | P(Y = y) = (1 − p)yp |
Probability Types
| Mode | For trials convention | Meaning |
|---|---|---|
| Exact | P(X = k) | The first success occurs exactly on trial k. |
| Less than | P(X < k) | The first success occurs before trial k. |
| At most | P(X ≤ k) | The first success occurs on or before trial k. |
| At least | P(X ≥ k) | The first success occurs on or after trial k. |
| More than | P(X > k) | The first success occurs after trial k. |
| Between | P(k ≤ X ≤ b) | The first success occurs between two trial numbers. |
Mean, Variance, and Standard Deviation
For the trial-number convention:
Variance = (1 − p) / p²
Standard Deviation = √(1 − p) / p
For the failures-before-success convention, the variance is the same, but the mean is:
Example: First Sale on a Call
Suppose a salesperson has a 25% chance of making a sale on each call. The probability that the first sale occurs on the 3rd call is:
p = 0.25 k = 3 P(X = 3) = (1 − 0.25)² × 0.25 P(X = 3) = 0.75² × 0.25 P(X = 3) = 0.140625
Example: First Defective Product
Be careful with the meaning of “success.” If you are calculating the first defective product, then success means “defective.” So if a machine has a 90% chance of producing a good product, the defect probability is 10%, so use p = 0.10 for first defect.
When This Distribution Makes Sense
- Each trial has only two outcomes: success or failure.
- The probability of success is constant.
- Trials are independent.
- You are interested in the first success.
Common Mistakes
- Using p = 90% for “first defective” when 90% is actually the good-product probability.
- Confusing “at least k” with “more than k.”
- Using the binomial distribution when the number of trials is not fixed.
- Using the geometric distribution when you need the second, third, or fourth success.
- Mixing the trial-number convention with the failures-before-success convention.
Frequently Asked Questions
What does the geometric distribution model?
It models the number of trials needed to get the first success, or the number of failures before the first success.
What is the formula for the first success on trial k?
P(X = k) = (1 − p)k−1p, where p is the probability of success on one trial.
What is the difference between geometric and binomial distributions?
The binomial distribution counts successes in a fixed number of trials. The geometric distribution counts how many trials are needed until the first success.
Can geometric distribution model multiple successes?
No. It models the first success. For multiple successes, the negative binomial distribution is more appropriate.
Can p be 1?
Yes. If p = 1, the first trial is always a success. If p = 0, a success never occurs, so this calculator requires p to be greater than 0.