What Is a Geometric Distribution?
The geometric distribution models the number of trials needed to achieve the first success in a sequence of independent Bernoulli trials. Each trial has the same probability of success p and probability of failure q = 1 - p. It is a discrete probability distribution commonly used in quality control, reliability engineering, and games of chance.
For example, if you flip a fair coin, the geometric distribution tells you the probability that you get your first heads on the 3rd flip. The distribution is memoryless, meaning the probability of success on the next trial is always the same regardless of how many failures have occurred.
Formulas
Where p is the probability of success on each trial and k is the trial number on which the first success occurs. The PMF gives the probability that exactly k trials are needed, while the CDF gives the probability that the first success occurs on or before the k-th trial.
Probability Table
| p | Mean | Variance | P(X=1) | P(X≤5) |
|---|---|---|---|---|
| 0.1 | 10.00 | 90.00 | 0.1000 | 0.4095 |
| 0.2 | 5.00 | 20.00 | 0.2000 | 0.6723 |
| 0.3 | 3.33 | 7.78 | 0.3000 | 0.8319 |
| 0.5 | 2.00 | 2.00 | 0.5000 | 0.9688 |
| 0.8 | 1.25 | 0.31 | 0.8000 | 0.9997 |
Worked Examples
Example 1: Coin Flip
What is the probability of getting the first heads on the 3rd flip of a fair coin? Here p = 0.5 and k = 3. P(X = 3) = (1 - 0.5)^2 x 0.5 = 0.25 x 0.5 = 0.125, or 12.5%.
Example 2: Manufacturing Defects
A factory produces items with a 5% defect rate. What is the probability the first defective item is found on the 10th inspection? Here p = 0.05 and k = 10. P(X = 10) = (0.95)^9 x 0.05 = 0.6302 x 0.05 = 0.0315, or about 3.15%.
Frequently Asked Questions
What is the difference between geometric and negative binomial distributions?
The geometric distribution is a special case of the negative binomial distribution where the number of successes r = 1. The negative binomial models the number of trials needed for r successes, while the geometric models just the first success.
Is the geometric distribution memoryless?
Yes, the geometric distribution is the only discrete memoryless distribution. This means P(X > s + t | X > s) = P(X > t). If you have already failed s times, the probability of needing at least t more trials remains unchanged.
When should I use the geometric distribution?
Use it when you want to find the probability of needing a certain number of independent trials to achieve the first success, such as the number of dice rolls to get a six, the number of sales calls before a sale, or the number of lottery tickets before a win.