Compound Probability Examples in Math

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

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 of two or more events occurring together ( $P(A \text{ and } B)$ ) or at least one occurring ( $P(A \text{ or } B)$ ), accounting for whether the events are independent or dependent.

Single-event probability asks about one thing happening. Compound probability asks about combinations: 'What's the chance of rolling a 6 AND flipping heads?' or 'What's the chance of drawing a heart OR a face card?' The word 'and' usually means multiply; the word 'or' usually means add (but subtract the overlap).

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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: Compound probability combines two or more events: $P(A\text{ and }B)$ multiplies, $P(A\text{ or }B)$ adds then subtracts the overlap.

Common stuck point: The procedure for compound probability is the easy part; the trap is adding for 'and' or multiplying for 'or'. Asking "Does the problem combine two or more events with 'and' (multiply) or 'or' (add minus overlap)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does the problem combine two or more events with 'and' (multiply) or 'or' (add minus overlap)?

Worked Examples

Example 1

medium

Events A and B:

P(A)=0.5

P(B)=0.4

P(A \cap B)=0.2

. Find (a)

P(A \cup B)

, (b)

P(A|B)

, and verify whether A and B are independent.

Answer

(a)

P(A \cup B) = 0.7

. (b)

P(A|B) = 0.5

. A and B are independent.

First step

(a) Addition rule:

P(A \cup B) = P(A)+P(B)-P(A \cap B) = 0.5+0.4-0.2 = 0.7

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Example 2

hard

A card is drawn from a standard deck. Event A: card is red. Event B: card is a face card (J, Q, K). Find

P(A \cup B)

and

P(A \cap B)

Example 3

easy

Two fair dice are rolled. Find

P(\text{both odd})

by listing or by the product rule.

Example 4

medium

A jar has 4 red, 3 green, 2 yellow. Draw 2 without replacement. Find

P(\text{both same color})

Example 5

medium

From a deck, draw 1 card. Find

P(\text{red or king})

Example 6

hard

A drawer has 6 socks: 4 black, 2 white. Pull 2 randomly. Find

P(\text{match})

Practice Problems

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

Example 1

easy

P(A)=0.3

P(B)=0.5

, and A and B are mutually exclusive. Find

P(A \cup B)

Example 2

hard

Using the law of total probability:

P(B) = P(B|A)P(A) + P(B|A^c)P(A^c)

, find

P(B)

given

P(A)=0.4

P(B|A)=0.7

P(B|A^c)=0.3

Example 3

easy

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

P(\text{heads and } 6)

Example 4

easy

You roll a fair die. Find

P(\text{rolling a 2 or a 5})

Example 5

easy

Two fair coins are flipped. Find

P(\text{both heads})

Example 6

easy

A bag has 3 red and 2 blue marbles. Draw one. Find

P(\text{red or blue})

Example 7

easy

A spinner has 4 equal sections numbered 1-4, spun twice. Find

P(\text{first is 1 and second is 1})

Example 8

easy

A card is drawn from a standard deck. Find

P(\text{heart or spade})

Example 9

easy

Two fair dice are rolled. Find

P(\text{first is even and second is odd})

Example 10

easy

A coin is flipped 3 times. Find

P(\text{all three tails})

Example 11

medium

A card is drawn from a standard deck. Find

P(\text{heart or face card})

. (There are 13 hearts, 12 face cards, and 3 face cards that are hearts.)

Example 12

medium

A bag has 4 red and 6 green marbles. Two are drawn without replacement. Find

P(\text{both red})

Example 13

medium

A bag has 4 red and 6 green marbles. Two are drawn with replacement. Find

P(\text{both red})

Example 14

medium

A student has

P(\text{pass math})=0.8

and

P(\text{pass science})=0.7

, independent. Find

P(\text{pass both})

Example 15

medium

Using the data above (

P(\text{math})=0.8

P(\text{science})=0.7

, independent), find

P(\text{pass at least one})

Example 16

medium

A spinner lands on red with probability

0.3

. It is spun twice. Find

P(\text{exactly one red})

Example 17

medium

Two dice are rolled. Find

P(\text{sum} = 7)

Example 18

medium

A box has 5 good and 1 defective bulb. Two are drawn without replacement. Find

P(\text{at least one defective})

Example 19

challenge

In a class,

60\%

play soccer,

30\%

play tennis, and

20\%

play both. Find

P(\text{a student plays neither})

Example 20

challenge

A jar has 3 red, 4 blue, 5 green marbles. Three are drawn without replacement. Find

P(\text{one of each color})

Example 21

challenge

Events

A

and

B

satisfy

P(A)=0.5

P(B)=0.4

, and

P(A\text{ or }B)=0.7

. Are

A

and

B

independent?

Example 22

medium

A bag has 5 red and 5 blue marbles. Two are drawn without replacement. Find

P(\text{one red and one blue, in any order})

Example 23

easy

A fair die is rolled twice. Find

P(\text{both rolls show a 4})

Example 24

easy

A coin is flipped and a die rolled. Find

P(\text{tails and an even number})

Example 25

easy

A bag has 7 red and 3 blue marbles. Find

P(\text{red})

on a single draw.

Example 26

medium

A bag has 6 red, 4 blue marbles. Draw 2 without replacement. Find

P(\text{both blue})

Example 27

medium

P(A)=0.6

P(B)=0.5

P(A\cap B)=0.2

. Find

P(A\cup B)

Example 28

medium

P(A)=0.4

P(B)=0.3

, independent. Find

P(A\cup B)

Example 29

medium

A card is drawn from a standard deck. Find

P(\text{king or queen})

Example 30

medium

A coin is flipped 4 times. Find

P(\text{exactly 2 heads})

Example 31

medium

A die is rolled 3 times. Find

P(\text{at least one 6})

Example 32

medium

Two cards are drawn without replacement from a deck. Find

P(\text{both aces})

Example 33

medium

A factory makes parts; 95% pass QA independently. Find

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

Example 34

medium

P(A)=0.7

P(A\cap B)=0.21

. Are

A,B

independent if

P(B)=0.3

Example 35

hard

A bag has 3 red, 3 blue, 4 white. Draw 3 without replacement. Find

P(\text{all different colors})

Example 36

hard

Three independent traffic lights have

P(\text{green})=0.4

each. Find

P(\text{exactly 2 green})

Example 37

hard

A test is 95% accurate. Disease prevalence is 1%. Given a positive test, find

P(\text{has disease})

Example 38

hard

Two dice are rolled. Find

P(\text{sum}\ge 10)

Example 39

hard

P(A)=0.6

P(B)=0.5

P(A\cup B)=0.8

. Find

P(B|A)

Example 40

challenge

5 people randomly sit in a row of 5 chairs. Find

P(\text{Alex and Bo sit next to each other})

Example 41

challenge

In a group of 23 people, find

P(\text{at least 2 share a birthday})

, ignoring leap years. (Express via the standard formula.)

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Related Concepts

Probability Independent Events Conditional Probability Binomial Distribution Expected Value

Background Knowledge

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

probabilityindependent eventsconditional probability