The Geometric Distribution

You’ll be able to

Recognize a geometric setting and contrast it with a binomial setting
Compute the probability that the first success occurs on trial k
Find the expected number of trials until the first success

Geometric: waiting for the first success

A geometric setting shares three of the binomial conditions — binary trials, independence, and the same success probability p — but instead of a fixed number of trials, we keep going until the first success. The random variable X is the number of trials needed to get that first success, so X can be 1, 2, 3, … with no fixed upper limit. "How many attempts until it works?" signals geometric; "how many successes in n attempts?" signals binomial.

The probabilities and the mean

To get the first success on trial k, the first k − 1 trials must all fail and the kth must succeed: P(X = k) = (1 − p)^(k−1) · p. The expected number of trials until the first success is μ_X = 1/p — intuitive, since if success happens 1 in every p of the time, you wait on average 1/p tries. (For p = 0.25 you expect 4 tries.)

Geometric probability and mean

P(X = k) = (1 − p)^(k−1) · p · μ_X = 1 / p · σ_X = sqrt(1 − p) / p

X is the trial on which the first success occurs. The k − 1 failures come first, each with probability (1 − p), then one success with probability p.

Worked example

A basketball player makes each free throw with probability 0.25. Find the probability her first make is on the 3rd attempt, and the expected number of attempts until her first make.

1.Confirm geometric: binary (make/miss), independent, same p = 0.25, counting trials until the first success.
2.First success on trial 3 means miss, miss, make: P(X = 3) = (0.75)^2 · (0.25).
3.Compute: (0.5625)(0.25) ≈ 0.141.
4.Expected trials: μ = 1/p = 1/0.25 = 4.

Answer: P(first make on 3rd attempt) = (0.75)²(0.25) ≈ 0.141, and the expected number of attempts until the first make is 1/0.25 = 4.

Checkpoint

A game is won with probability 0.1 each time it is played, independently. What is the expected number of games until the first win?

On the exam

The quickest way to tell binomial from geometric: is the number of trials fixed? Fixed n, count successes → binomial. Keep going until the first success, count the trials → geometric. The phrase "until" almost always means geometric.

Checkpoint

A quality checker inspects items until the first defective one appears; 5% of items are defective. What is the probability the first defective is the 4th item inspected?

Answer the 2 checkpoints as you read.