Independent Events Examples in Statistics

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

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

Concept Recap

Two events are independent if knowing that one event happened does not change the probability of the other event.

Independence means “no update.” If learning B happened leaves the chance of A exactly the same, then the events are independent.

Read the full concept explanation →

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: Independent does not mean “separate topics.” It means one event gives no probabilistic information about the other.

Common stuck point: Students often assume events are independent just because the story describes two different actions. Recognition is harder than the formula.

Sense of Study hint: Ask: after I learn one event happened, does the probability of the other event stay the same or change?

Common Mistakes to Watch For

Before you work through the examples, skim the mistake guide so you know which shortcuts and sign errors to avoid.

Conditional Probability Mistakes

Worked Examples

Example 1

medium

From sample data,

P(A)=0.4

P(B)=0.5

P(A\cap B)=0.2

. Are

A

and

B

independent?

Answer

\text{Yes — } P(A)\,P(B)=0.20=P(A\cap B)

First step

Check

P(A)\cdot P(B)=0.4\cdot 0.5=0.20

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Example 2

medium

P(A)=0.5

P(B \mid A)=0.3

, and

P(B)=0.3

. Are

A

and

B

independent? Justify.

Example 3

medium

A sample is collected by Bernoulli sampling with

P(\text{included})=0.10

. Three independent units are considered. Find

P(\text{exactly 1 included})

Example 4

hard

An A/B test runs until either A or B gets 3 conversions first. Assignments are independent and

P(\text{convert} \mid A)=P(\text{convert} \mid B)=0.5

. Find

P(\text{A gets exactly 3 conversions in the first 5 trials})

when only A's group is tracked.

Example 5

hard

Two cards are drawn from a 52-card deck without replacement. Show whether

P(\text{2nd is ace})

is affected by knowing the first was an ace.

Example 6

challenge

Suppose

A

and

B

are independent with

P(A)=0.4

. We sample data and find

P(B \mid A)=0.30

. Is the sample evidence consistent with independence? What is

P(B)

under independence?

Example 7

easy

A red die and a blue die are rolled together. Find

P(\text{red is }3\text{ and blue is even})

Red die → Blue die; highlighted path Red=3→Even has probability 1/6 × 1/2 = 1/12

Example 8

medium

A free-throw shooter makes

80\%

of shots. Find

P(\text{makes 4 in a row})

, assuming independence.

Example 9

medium

Two independent components have failure probabilities

0.05

and

0.10

. Find

P(\text{both work})

Example 10

medium

A circuit has two switches in series, each open with probability

0.2

independently. Find

P(\text{circuit closed})

Example 11

medium

A spinner with

P(\text{red}) = 0.25

is spun

3

times independently. Find

P(\text{exactly }1\text{ red})

Example 12

medium

A die is rolled and a coin is flipped. Let

A = \{\text{die is }6\}

and

B = \{\text{coin is heads}\}

. Show

A,B

are independent.

Example 13

hard

Three independent events

A, B, C

each have probability

0.6

. Find

P(\text{exactly }2\text{ occur})

Example 14

hard

A system has

4

independent components, each working with probability

0.9

. The system works if at least

3

work. Find

P(\text{system works})

Example 15

hard

Show that if

A,B

are independent, then

A

and

B^c

are also independent.

Example 16

challenge

A fair coin is flipped

10

times. Find

P(\text{no two consecutive heads})

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 twice. Find

P(\text{heads then heads})

Flip 1 → Flip 2; highlighted path H→H has probability 1/2 × 1/2 = 1/4

Example 2

easy

A die is rolled and a coin is flipped. Find

P(\text{6 and heads})

Die roll → Coin flip; highlighted path 6→H has probability 1/6 × 1/2 = 1/12

Example 3

easy

Events

A

and

B

are independent with

P(A)=0.3

and

P(B)=0.5

. Find

P(A\cap B)

Example 4

easy

Two independent events each have probability

0.4

. Find the probability both occur.

Example 5

easy

A bag has 10 balls. You draw one, replace it, then draw again. Are the two draws independent?

Example 6

easy

A spinner lands on red with probability

0.2

. Spun twice independently, find

P(\text{red both times})

Spin 1 → Spin 2; highlighted path Red→Red has probability 0.2 × 0.2 = 0.04

Example 7

easy

P(A)=0.5

P(B)=0.6

, and

P(A\cap B)=0.3

, are

A

and

B

independent?

Example 8

easy

Two events are mutually exclusive with nonzero probabilities. Can they be independent?

Example 9

medium

A student passes math with probability

0.8

and passes science with probability

0.9

, independently. Find the probability of passing both.

Math result → Science result; highlighted path Pass M→Pass S has probability 0.8 × 0.9 = 0.72

Example 10

medium

Using the previous setup (

0.8

math,

0.9

science, independent), find the probability of passing at least one subject.

Math → Science; Fail M→Fail S has probability 0.2 × 0.1 = 0.02, so P(at least one pass) = 1 − 0.02 = 0.98

Example 11

medium

A machine has two independent components, each failing with probability

0.1

. Find the probability the machine works (needs both components working).

Component 1 → Component 2; Works→Works has probability 0.9 × 0.9 = 0.81

Example 12

medium

Three independent traffic lights are green with probability

0.6

each. Find the probability all three are green.

Light 1 → Light 2 → Light 3; highlighted path Green→Green→Green has probability 0.6³ = 0.216

Example 13

medium

Events

A

and

B

are independent with

P(A)=0.4

P(B)=0.7

. Find

P(A\cup B)

Example 14

medium

A test for a disease is positive with probability

0.95

if a person is sick. Two independent tests are run on a sick person. Find

P(\text{both positive})

Test 1 → Test 2; highlighted path Pos→Pos has probability 0.95 × 0.95 = 0.9025

Example 15

medium

A coin is flipped 4 times independently. Find the probability of getting all tails.

Example 16

medium

P(A)=0.5

A

and

B

are independent and

P(A\cap B)=0.2

. Find

P(B)

Example 17

medium

Two archers hit a target independently with probabilities

0.7

and

0.8

. Find the probability that exactly one hits.

Archer A → Archer B; exactly one hit = P(Hit A, Miss B) + P(Miss A, Hit B) = 0.14 + 0.24 = 0.38

Example 18

challenge

Events

A

and

B

are independent. Prove that

A

and

B^c

are also independent.

Example 19

challenge

A system works if at least one of three independent backups works. Each works with probability

0.6

. Find the probability the system works.

Example 20

challenge

For independent events with

P(A)=p

and

P(B)=p

, the probability of at least one is

0.75

. Find

p

Example 21

easy

A random sample draws two subjects with replacement. If

P(\text{female})=0.50

, find the probability both are female.

Draw 1 → Draw 2 (with replacement); highlighted path Female→Female has probability 0.50 × 0.50 = 0.25

Example 22

easy

Two independent diagnostic tests each have

P(\text{positive} \mid \text{healthy})=0.05

. Find

P(\text{both false positive})

Example 23

easy

A coin and a die are observed as independent trials. Find

P(\text{heads and 4 or higher})

Example 24

easy

A simple random sample of 3 households is drawn with replacement. If

P(\text{owns car})=0.80

, find

P(\text{all 3 own a car})

Household 1 → 2 → 3 (with replacement); Car→Car→Car has probability 0.8³ = 0.512

Example 25

medium

In a Bernoulli sampling design, each subject independently has

P(\text{respond})=0.6

. Find

P(\text{all 4 respond})

Example 26

medium

A sensor's two independent readings each fail with probability

0.02

. Find

P(\text{at least one fails})

Sensor 1 → Sensor 2; P(both OK) = 0.98² = 0.9604; P(at least one fails) = 1 − 0.9604 = 0.0396

Example 27

medium

Two independent random samples have respondent rates

0.7

and

0.6

. Find

P(\text{a respondent from each})

Example 28

medium

A clinical trial randomly assigns each patient independently to treatment with

P=0.5

. Find

P(\text{exactly 2 of 3 patients are assigned to treatment})

Example 29

medium

Two independent surveys each have a 5% non-response rate. Find

P(\text{at least one survey gets a non-response from a sampled subject})

, assuming a subject responds independently in each.

Example 30

medium

Five independent measurements of a sample each are within tolerance with probability

0.9

. Find

P(\text{all 5 within tolerance})

Example 31

medium

P(A)=0.6

P(B)=0.3

A

and

B

are independent. Find

P(A\cup B)

Example 32

medium

Two independent normal random variables each have

P(X>0)=0.5

. Find

P(\text{both positive})

Example 33

medium

A randomized A/B test assigns each visitor to A with

P=0.5

independently. Of 4 visitors, find

P(\text{all assigned to A})

Example 34

medium

Two independent estimators each give

P(\text{within 1 SE})\approx 0.68

. Find

P(\text{both within 1 SE})

Example 35

hard

Three independent hypothesis tests, each with significance level

\alpha=0.05

, are run on null-true data. Find

P(\text{at least one false rejection})

Example 36

hard

Two independent confidence intervals each cover the true parameter with probability

0.95

. Find

P(\text{both cover})

Example 37

hard

From sampled census tracts,

P(\text{rural})=0.30

. Three tracts are sampled independently. Find

P(\text{at least 2 rural})

Example 38

hard

P(A)=0.6

P(B)=0.5

. If

A,B

were independent, what would

P(A\cap B)

be? Observed

P(A\cap B)=0.40

; are they independent?

Example 39

hard

A study independently samples 4 subjects; each has

P(\text{respond})=0.8

. Find

P(\text{at least 3 respond})

Example 40

hard

A simple random sample of 5 voters has

P(\text{vote yes})=0.55

for each (with replacement). Find

P(\text{at least one no})

Example 41

easy

A coin is flipped three times. Find

P(\text{HHH})

Flip 1 → Flip 2 → Flip 3; highlighted path H→H→H has probability (1/2)³ = 1/8

Example 42

easy

A die is rolled twice. Find

P(\text{6 then 6})

Roll 1 → Roll 2; highlighted path 6→6 has probability 1/6 × 1/6 = 1/36

Example 43

easy

A bag has

5

balls. Draw one WITHOUT replacement, then another. Are the draws independent?

Example 44

easy

Events

A,B

are independent. If

P(A) = 0

what is

P(A\cap B)

Example 45

easy

A

and

B

are independent with

P(A) = 0.3

and

P(B) = 0.4

. Find

P(A\mid B)

Example 46

medium

A test has

4

true/false questions. If a student guesses all, find

P(\text{all correct})

Example 47

medium

Two independent events have

P(A) = 0.6

P(B) = 0.5

. Find

P(A\cup B)

Example 48

medium

A weather model gives

P(\text{rain today}) = 0.3

and

P(\text{rain tomorrow}) = 0.4

, independent. Find

P(\text{rain on at least one day})

Today → Tomorrow; P(no rain both) = 0.7 × 0.6 = 0.42; P(rain at least one day) = 1 − 0.42 = 0.58

Example 49

medium

A bag has 3 red and 7 blue. Draw with replacement twice. Find

P(\text{red then blue})

Draw 1 → Draw 2 (with replacement); highlighted path Red→Blue has probability 3/10 × 7/10 = 21/100

Example 50

hard

A fair coin is flipped until heads appears. Find

P(\text{exactly }3\text{ flips needed})

Example 51

hard

Two independent events:

P(A) = 0.5

P(B) = x

P(A\cup B) = 0.7

. Find

x

Example 52

hard

A coin is flipped

5

times. Find

P(\text{exactly }3\text{ heads})

Example 53

hard

Two independent shooters hit a target with probabilities

0.7

and

0.4

. They each take one shot. Find

P(\text{exactly one hit})

Example 54

challenge

A, B, C

are pairwise independent, each with probability

1/2

, but

P(A\cap B\cap C) = 1/4

, not

1/8

. Are

A, B, C

mutually independent?

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Related Concepts

Conditional Probability Multiplication Rule Expected Value

Background Knowledge

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

conditional probability