Independent Events Examples in Math

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 Math.

Concept Recap

Two events are independent if the occurrence of one does not change the probability of the other: $P(A \cap B) = P(A) \cdot P(B)$ .

They don't 'know about' each other. One happening tells you nothing about the other.

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: Two events are independent when knowing one happened does not change the chance of the other.

Common stuck point: The procedure for independent events is the easy part; the trap is multiplying probabilities for without-replacement draws. Asking "Does the first event happening change the probability of the second?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does the first event happening change the probability of the second?

Worked Examples

Example 1

easy

A fair coin is flipped and a fair die is rolled. Find

P(\text{Heads and a 4})

Answer

P(\text{Heads and a 4}) = \frac{1}{12}

First step

Identify the two events:

A = \text{Heads}

B = \text{rolling a 4}

Full solution

2
Check independence: the coin flip does not affect the die roll, so $A$ and $B$ are independent
3
Find individual probabilities: $P(A) = \frac{1}{2}$ , $P(B) = \frac{1}{6}$
4
Apply multiplication rule: $P(A \cap B) = P(A) \cdot P(B) = \frac{1}{2} \cdot \frac{1}{6} = \frac{1}{12}$

Two events are independent if the occurrence of one does not change the probability of the other. For independent events,

P(A \cap B) = P(A) \cdot P(B)

. Physical separation of experiments is the clearest indicator of independence.

Example 2

medium

For two events

A

and

B

P(A) = 0.4

and

P(B) = 0.5

. If

A

and

B

are independent, find (a)

P(A \cap B)

and (b)

P(A \cup B)

Example 3

medium

A coin is flipped and a die is rolled. Find P(heads AND rolling a 6).

Example 4

medium

A weather model says

P(\text{rain on Saturday})=0.4

and

P(\text{rain on Sunday})=0.3

, treated as independent. Find

P(\text{rain both days})

and

P(\text{no rain either day})

Example 5

medium

Two fair dice are rolled. Let

A

= "first die is even" and

B

= "sum is

7

." Are

A

and

B

independent?

Example 6

medium

A light bulb has probability

0.02

of failing in a year, independent of others. A fixture uses

3

bulbs. Find

P(\text{at least one fails})

Example 7

hard

P(A)=0.5

P(B)=0.4

, and

P(A \cup B)=0.7

. Are

A

and

B

independent?

Example 8

hard

A test for a virus has

98\%

sensitivity. If three independent tests are run on a true positive, find

P(\text{all three tests positive})

and

P(\text{at least one test misses})

Example 9

hard

Two independent events satisfy

P(A)=p

P(B)=2p

, and

P(A \cap B)=0.18

. Find

p

Example 10

challenge

A device works if at least

2

3

independent components work, each with probability

0.9

. Find the probability the device works.

Example 11

hard

Show that if

A,B

are independent, then

A

and

B^c

are also independent.

Example 12

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 student randomly guesses on two multiple-choice questions, each with 4 options. Find the probability of getting both correct.

Example 2

hard

Events

A

and

B

satisfy

P(A) = 0.6

P(B) = 0.7

, and

P(A \cap B) = 0.42

. Are

A

and

B

independent? Justify algebraically.

Example 3

easy

Are two flips of a fair coin independent?

Example 4

easy

P(A)=0.5

P(B)=0.4

, independent. Find

P(A\cap B)

Example 5

easy

Two independent events:

P(A)=0.3

P(B)=0.6

. Find

P(A\cap B)

Example 6

easy

Rolling a die and flipping a coin: independent?

Example 7

easy

P(two independent events both occur) when each has probability

0.5

Example 8

easy

Drawing a card, replacing it, then drawing again: are the draws independent?

Example 9

easy

Does 'mutually exclusive' mean 'independent'?

Example 10

easy

Independent events

A,B

each probability

0.2

. P(neither occurs)?

Example 11

medium

P(A)=0.4

P(B)=0.5

, independent. Find

P(A\cup B)

Example 12

medium

A machine works if both independent parts work, each with reliability

0.9

. P(machine works)?

Example 13

medium

Test independence:

P(A)=0.3

P(B)=0.5

P(A\cap B)=0.2

. Independent?

Example 14

medium

A coin is flipped 4 times. P(all heads)?

Example 15

medium

Two dice: P(both show 6)?

Example 16

medium

P(at least one head in 3 independent flips)?

Example 17

medium

Independent events

A,B

P(A)=0.6

P(A\cap B)=0.24

. Find

P(B)

Example 18

medium

A free throw shooter makes 80% independently. P(makes first two, misses third)?

Example 19

medium

Independent events

A,B

P(A)=0.5

P(B)=0.2

. P(exactly one occurs)?

Example 20

challenge

A

and

B

are independent, show

A

and

B'

(complement) are also independent for

P(A)=0.5

P(B)=0.4

Example 21

challenge

Three independent components each fail with probability

0.1

. P(at least one fails)?

Example 22

challenge

P(A)=p

P(B)=0.5

, independent, and

P(A\cup B)=0.7

. Find

p

Example 23

easy

A fair coin is flipped twice. Find

P(\text{Heads on both flips})

Example 24

easy

A spinner with

5

equal sectors and a fair coin are spun and flipped together. Find

P(\text{sector } 1 \text{ and Tails})

Example 25

easy

Two fair dice are rolled. Find

P(\text{first die shows } 2 \text{ and second die shows } 5)

Example 26

easy

P(A)=0.2

and

P(B)=0.5

are independent. Find

P(A \cap B)

Example 27

easy

A bag has

3

red and

2

blue marbles. You draw one, replace it, then draw again. Find

P(\text{both red})

Example 28

medium

A free-throw shooter makes

80\%

of attempts. Assuming shots are independent, find

P(\text{makes all 3 of 3})

Example 29

medium

Two independent events satisfy

P(A)=0.4

and

P(B)=0.5

. Find

P(\text{exactly one occurs})

Example 30

medium

A bag has

4

red and

6

blue marbles. Two marbles are drawn with replacement. Find

P(\text{first red and second blue})

Example 31

medium

A factory has two independent quality checks, each catching defects with probability

0.9

. Find

P(\text{at least one check catches a given defect})

Example 32

medium

You roll a die and toss a coin. Find

P(\text{die shows a prime number AND coin shows Heads})

Example 33

hard

A basketball player makes

70\%

of free throws. Assuming independence, find

P(\text{makes exactly 2 of 3 attempts})

Example 34

hard

Two events satisfy

P(A)=0.6

P(B)=0.5

, and

P(A \cup B)=0.9

. Are they independent?

Example 35

hard

A circuit has three independent switches, each closed with probability

0.9

. Current flows only if ALL three are closed. Find

P(\text{current flows})

Example 36

hard

A coin is flipped

5

times. Find

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

Example 37

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 38

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?

← Back to Independent Events Formula Explained Practice Mode

Related Concepts

Probability Dependence (Statistical)Conditional Probability

Background Knowledge

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

probability