Geometric Probability
Probability that the first success occurs on trial k.
Formula
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Example
First head on 3rd flip: (0.5)²(0.5) = 0.125
How it works
This formula gives the probability that the first success happens exactly on trial k when each trial is independent and has the same success probability p. To get that outcome, the first k-1 trials must all be failures, which contributes (1-p)^(k-1), and then the k-th trial must be a success, which contributes p. Multiplying these independent parts gives the total probability.
When to use it
Use this when you are counting repeated yes/no trials until the first success occurs, such as the first defective item found, the first customer who buys, or the first shot made in sports. It applies only when the success chance stays constant from trial to trial and each trial is independent.
Worked example, step by step
Suppose a basketball player makes a free throw with probability 0.7 on each attempt. We want the probability that the first made shot occurs on the 4th attempt. Use P(X = k) = (1 - p)^(k - 1) p P(X = 4) = (1 - 0.7)^(4 - 1)(0.7) P(X = 4) = (0.3)^3(0.7) P(X = 4) = 0.027 × 0.7 P(X = 4) = 0.0189 So, the probability is 0.0189, or 1.89%.
Common mistakes to avoid
- Forgetting that the first k-1 trials must all be failures before the success
- Using k instead of k-1 in the exponent
- Applying the formula when the trial probabilities change from one attempt to the next
Did You Know?
The geometric distribution is memoryless.
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