Practice Geometric Distribution in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

The probability distribution for the number of independent Bernoulli trials needed to get the first success, where each trial has success probability pp.

How many times do you have to roll a die before you get a 6? The geometric distribution answers this kind of question. Each trial is independent, and you keep going until you succeed. Most of the time it doesn't take too long, but occasionally you have an unlucky streak—that's why the distribution has a long right tail.

Showing a random 20 of 50 problems.

Example 1

hard
Use the geometric series to verify that the PMF P(X=k)=(1−p)k−1pP(X = k) = (1-p)^{k-1} p sums to 11 over k=1,2,3,…k = 1, 2, 3, \ldots.

Example 2

medium
A basketball player makes free throws with probability p=0.7p=0.7. Find the probability they make their first free throw on exactly the 3rd attempt.

Example 3

easy
A salesperson closes each call with probability p=0.25p = 0.25. What is the expected number of calls to the first sale?

Example 4

medium
Given XX is geometric with p=0.2p = 0.2 and we know X>3X > 3, what is P(X>7∣X>3)P(X > 7 \mid X > 3)?

Example 5

easy
Rolling a die until a 66 appears, p=16p = \tfrac{1}{6}. What is the expected number of rolls?

Example 6

medium
A player believes 'I've missed 1010 times, so I'm due for a hit.' What property of the geometric distribution refutes this?

Example 7

challenge
Derive why the geometric mean is 1/p1/p from the tail-sum identity E[X]=∑k=0∞P(X>k)=∑k=0∞(1−p)kE[X] = \sum_{k=0}^{\infty} P(X > k) = \sum_{k=0}^{\infty}(1-p)^k.

Example 8

medium
A die is rolled until a 55 appears. Find P(X≤2)P(X \le 2).

Example 9

medium
With p=0.1p = 0.1, what is the expected number of trials until the first success, and what does it imply about typical waits?

Example 10

easy
A spinner lands on red with probability 0.40.4. Find P(first red on spin 2)P(\text{first red on spin } 2).

Example 11

medium
Why is the binomial distribution not appropriate when modeling 'number of attempts until the first success'?

Example 12

easy
For a geometric distribution with success probability pp, what is P(X=1)P(X = 1) (success on the first trial)?

Example 13

easy
Using P(X=k)=(1−p)k−1pP(X = k) = (1-p)^{k-1}p with p=0.2p = 0.2, find P(X=2)P(X = 2).

Example 14

medium
A geometric random variable has p=0.25p = 0.25. Find P(X>4)P(X > 4).

Example 15

medium
With p=0.5p = 0.5, find P(X=4)P(X = 4) using P(X=k)=(1−p)k−1pP(X=k) = (1-p)^{k-1}p.

Example 16

medium
If XX is geometric with p=0.4p = 0.4, compute Var(X)\text{Var}(X).

Example 17

medium
A geometric RV with p=0.3p = 0.3. Compute P(2≤X≤4)P(2 \le X \le 4).

Example 18

medium
For a geometric variable with pp, what value of kk maximizes P(X=k)P(X = k)?

Example 19

medium
With p=0.5p = 0.5, find P(X≤2)P(X \le 2) (first success within the first two trials).

Example 20

easy
Fill in the blank: For a geometric distribution, the variance is given by Var(X)=____p2\text{Var}(X) = \frac{\_\_\_\_}{p^2}.
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