Geometric Distribution Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Geometric Distribution.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

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

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.

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How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: The geometric distribution gives the probability that the first success occurs on trial $k$ , when independent trials each succeed with probability $p$ .

Common stuck point: The procedure for geometric distribution is the easy part; the trap is using the binomial because both involve $p$ and $1-p$ . Asking "Am I counting the number of trials up to and including the first success (not the number of successes in a fixed set of trials)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I counting the number of trials up to and including the first success (not the number of successes in a fixed set of trials)?

Worked Examples

Example 1

medium

A basketball player makes free throws with probability

p=0.7

. Find the probability they make their first free throw on exactly the 3rd attempt.

Answer

P(X=3) = (0.3)^2(0.7) = 0.063

. 6.3% chance first make is on attempt 3.

First step

Geometric distribution:

P(X=k) = (1-p)^{k-1} \cdot p

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Example 2

hard

For a geometric distribution with

p=0.4

: (a) find

P(X \leq 3)

, (b) find the expected number of trials until first success.

Example 3

medium

Derive a closed form for

P(X \le n)

when

X

is geometric with parameter

p

Example 4

medium

Show that the geometric distribution is memoryless:

P(X > m + n \mid X > m) = P(X > n)

Example 5

hard

Use the geometric series to verify that the PMF

P(X = k) = (1-p)^{k-1} p

sums to

1

over

k = 1, 2, 3, \ldots

Example 6

hard

Find the moment generating function of a geometric random variable with parameter

p

Example 7

medium

A casino game pays out on

5\%

of plays. A gambler plans to play until winning once. Find

P(\text{wins within first 20 plays})

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

A fair coin is flipped until heads appears. What is

P(\text{first heads on flip 4})

, and what is the expected number of flips?

Example 2

hard

A quality inspector samples items until finding the first defective. Defect probability is

p=0.05

. Find

P(X > 10)

(probability of needing more than 10 inspections) and the expected number to inspect.

Example 3

easy

For a geometric distribution with success probability

p

, what is

P(X = 1)

(success on the first trial)?

Example 4

easy

Rolling a die until a

6

appears,

p = \tfrac{1}{6}

. What is the expected number of rolls?

Example 5

easy

A fair coin is flipped until heads,

p = 0.5

. What is the expected number of flips?

Example 6

easy

Using

P(X = k) = (1-p)^{k-1}p

with

p = 0.2

, find

P(X = 2)

Example 7

easy

In a geometric setting, is the number of trials fixed in advance?

Example 8

easy

True or false: each trial in a geometric distribution is independent with the same success probability

p

Example 9

easy

P(X = k) = (1-p)^{k-1}p

counts trials until first success, what is the smallest value

k

can take?

Example 10

easy

A salesperson closes each call with probability

p = 0.25

. What is the expected number of calls to the first sale?

Example 11

medium

With

p = 0.3

, find

P(X = 3)

using

P(X=k) = (1-p)^{k-1}p

Example 12

medium

With

p = 0.2

, find

P(X > 2)

, the probability the first success comes after trial

2

Example 13

medium

With

p = 0.5

, find

P(X \le 2)

(first success within the first two trials).

Example 14

medium

A free throw is made with probability

p = 0.8

. What is the probability the first miss occurs on the third attempt?

Example 15

medium

A player believes 'I've missed

10

times, so I'm due for a hit.' What property of the geometric distribution refutes this?

Example 16

medium

Should you use the binomial or geometric distribution to model 'number of trials until the first defective item'?

Example 17

medium

With

p = 0.1

, what is the expected number of trials until the first success, and what does it imply about typical waits?

Example 18

challenge

Derive why the geometric mean is

1/p

from the tail-sum identity

E[X] = \sum_{k=0}^{\infty} P(X > k) = \sum_{k=0}^{\infty}(1-p)^k

Example 19

challenge

Prove the memoryless property: show

P(X > m + n \mid X > m) = P(X > n)

for the geometric distribution.

Example 20

challenge

With

p = 0.25

, find the smallest number of trials

n

so that the probability of at least one success is at least

0.9

Example 21

medium

With

p = 0.4

, find

P(X > 3)

, the probability the first success comes after trial

3

Example 22

medium

With

p = 0.5

, find

P(X = 4)

using

P(X=k) = (1-p)^{k-1}p

Example 23

easy

A geometric random variable has

p = 0.3

. Compute

P(X = 1)

Example 24

easy

A free-throw shooter makes 80% of attempts. What is the probability the first miss happens on the 2nd shot?

Example 25

medium

A die is rolled until a

5

appears. Find

P(X \le 2)

Example 26

medium

A geometric random variable has

p = 0.25

. Find

P(X > 4)

Example 27

medium

X

is geometric with

p = 0.4

, compute

\text{Var}(X)

Example 28

medium

A computer reboot has a 10% chance of failure each attempt. What is the probability the first failure occurs on or before the 5th reboot?

Example 29

medium

Given

X

is geometric with

p = 0.2

and we know

X > 3

, what is

P(X > 7 \mid X > 3)

Example 30

hard

A lottery has win probability

p = 0.01

. What is the smallest

n

such that

P(X \le n) \ge 0.5

Example 31

hard

A fair die is rolled until a

1

appears. Given the first

1

has not happened in the first

5

rolls, what is the expected number of additional rolls?

Example 32

hard

A virus test has sensitivity

0.95

(true-positive rate). Assuming independent tests on a truly infected person, what is the probability the test misses a positive 3 times in a row before catching it on the 4th test?

Example 33

easy

A spinner lands on red with probability

0.4

. Find

P(\text{first red on spin } 2)

Example 34

medium

A geometric RV has

P(X = 1) = 0.6

. Find

P(X = 3)

Example 35

hard

Two geometric variables

X_1, X_2

each have

p = 0.5

and are independent. Find

P(X_1 + X_2 = 4)

Example 36

medium

A geometric RV with

p = 0.3

. Compute

P(2 \le X \le 4)

Example 37

challenge

A coupon machine dispenses a special golden coupon with probability

1/n

. Approximately how many coupons must be drawn so that the probability of having drawn at least one golden coupon exceeds

1 - 1/e

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Related Concepts

Binomial Distribution Independent Events Expected Value

Background Knowledge

These ideas may be useful before you work through the harder examples.

binomial distributionindependent eventsexpected value