Question 1

What is the geometric distribution and what does it model?

Accepted Answer

The geometric distribution models the number of independent Bernoulli trials needed until the first success. Each trial has two outcomes (success/failure) with a fixed success probability p, and trials are independent. Examples: number of sales calls until closing a deal; number of free throws until missing one; number of website visits until a purchase; number of times you roll a die until getting a 6 (p = 1/6).

Question 2

What is the mean and variance of the geometric distribution?

Accepted Answer

Mean (expected number of trials until first success) = 1/p. For p = 0.5 (coin flip), expect 2 trials; for p = 0.1 (10% success rate), expect 10 trials. Variance = (1−p)/p². Standard deviation = √((1−p)/p²) = √(1−p)/p. Note: as p → 0 (very rare success), the mean and variance both grow very large — you may need many trials before the first success.

Question 3

What is the memoryless property of the geometric distribution?

Accepted Answer

The geometric distribution is the only discrete distribution with the memoryless property: P(X > m + n | X > m) = P(X > n). In words: if you have already failed m times, the probability of needing at least n more trials is the same as if you were starting fresh. Each trial is independent — past failures provide no information about future outcomes. This is why 'I've been waiting so long, I must be due for a success' is a fallacy.

Question 4

What is the difference between the two versions of the geometric distribution?

Accepted Answer

There are two common parameterisations: (1) X = number of trials until first success (X ≥ 1) — used here: P(X=k) = (1−p)^(k−1) × p, mean = 1/p. (2) Y = number of failures before first success (Y ≥ 0): P(Y=k) = (1−p)^k × p, mean = (1−p)/p. The relationship is Y = X − 1. Be careful when reading formulas to check which convention is used — the two versions have different formulas for P(X=1) and different means.

Question 5

How is the geometric distribution related to the negative binomial?

Accepted Answer

The geometric distribution is a special case of the negative binomial distribution with r = 1 (waiting for the first success). The negative binomial with parameter r models the number of trials needed until r successes — generalising geometric to multiple successes. The sum of r independent geometric(p) random variables follows a negative binomial(r, p) distribution.

Question 6

When should I use geometric vs binomial distribution?

Accepted Answer

Use binomial when n (number of trials) is fixed and you ask "how many successes in n trials?" Use geometric when the number of trials is not fixed and you ask "how many trials until the first success?" Key difference: binomial has a fixed n and variable k (number of successes); geometric has a fixed goal (1 success) and variable k (number of trials needed). Binomial counts successes in a fixed experiment; geometric counts trials until a success.

Question type	Distribution	What is fixed
How many trials until first success?	Geometric (this calculator)	p; k is the unknown
How many successes in n trials?	Binomial	n and p; k (successes) is the unknown
How many trials until r-th success?	Negative Binomial	r and p; n (trials) is the unknown
Time until first event (continuous)	Exponential	Rate λ; t is continuous
Number of events in fixed time	Poisson	λ (rate × time)
Sampling without replacement	Hypergeometric	Population composition

Geometric Distribution Calculator

Formula

Geometric vs Binomial — Which to Use?

Case Study: Sales Call Success Rate

Related Statistics Tools

Related Calculators

Disclaimer

Frequently Asked Questions