Geometric Distribution Calculator
Calculate the probability that the first success occurs on the k-th trial in a sequence of independent Bernoulli trials.
P(X = k)
0.08192000
P(X = k) vs Success Probability (p)
Formula
## How to Calculate Geometric Distribution Probability ### Formula **P(X = k) = (1-p)^(k-1) * p** The geometric distribution models the number of trials needed to get the first success. Each trial is independent with success probability p. The probability of needing exactly k trials means k-1 failures followed by 1 success. The expected number of trials is 1/p.
Esempio Risolto
A basketball player has a 20% free-throw rate. What is the probability the first basket is made on the 5th attempt?
- 01P(X=5) = (1-0.2)^(5-1) * 0.2
- 02= 0.8^4 * 0.2
- 03= 0.4096 * 0.2
- 04= 0.08192
- 05Expected trials until first success = 1/0.2 = 5
Domande Frequenti
Is the geometric distribution memoryless?
Yes, it is the discrete analog of the exponential distribution and has the memoryless property: P(X > s+t | X > s) = P(X > t). Past failures do not affect future success probability.
What is the difference between the two versions of the geometric distribution?
One version counts trials until success (starting at k=1), and the other counts failures before success (starting at k=0). We use the "trials until success" version where P(X=k) = (1-p)^(k-1) * p.
How does geometric relate to negative binomial?
The geometric distribution is a special case of the negative binomial with r=1 (waiting for the first success). The negative binomial generalizes to waiting for the r-th success.
Impara