Practice Independent Events in Statistics

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

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Quick 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.

Showing a random 20 of 80 problems.

Example 1

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 2

hard

A,B

are independent.

P(A\cup B) = 0.7

and

P(B) = 0.5

. Find

P(A)

Example 3

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 4

hard

A coin is flipped

5

times. Find

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

Example 5

easy

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

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

Example 6

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 7

easy

Two independent diagnostic tests each have

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

. Find

P(\text{both false positive})

Example 8

easy

A spinner gives red with probability

0.3

. Spun twice independently, find

P(\text{red on both spins})

Spin 1 → Spin 2; highlighted path Red→Red has probability 0.3 × 0.3 = 0.09

Example 9

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 10

hard

A fair coin is flipped until heads appears. Find

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

Example 11

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 12

easy

Events

A,B

are independent. If

P(A) = 0

what is

P(A\cap B)

Example 13

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 14

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 15

medium

Two independent events have

P(A) = 0.6

P(B) = 0.5

. Find

P(A\cup B)

Example 16

hard

Three independent events

A, B, C

each have probability

0.6

. Find

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

Example 17

medium

P(A) = 0.4

P(B) = 0.5

P(A\cap B) = 0.25

. Are

A,B

independent?

Example 18

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 19

easy

A bag has

5

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

Example 20

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

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Conditional Probability Multiplication Rule Expected Value