Basic Probability Examples in Statistics

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

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

Concept Recap

Probability is the measure of how likely an event is to occur, expressed as a number between 0 (impossible) and 1 (certain). It is calculated as the ratio of favorable outcomes to total possible outcomes when all outcomes are equally likely.

Probability is a way of putting a number on chance. Flipping heads? That's $0.5$ (half the time). Rolling a 6 on a die? That's $\frac{1}{6}$ (one out of six possible outcomes). It's like asking 'if we did this many times, what fraction would this outcome happen?'

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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: Basic Probability starts by naming the possible outcomes and the event rule before assigning or combining probabilities.

Common stuck point: Students often know a procedure related to basic probability but skip the recognition step: Am I reasoning about what can happen and how likely it is, with the correct sample space or condition? That leads to a calculation or graph that looks reasonable but answers a different question.

Sense of Study hint: Ask: Am I reasoning about what can happen and how likely it is, with the correct sample space or condition?

Worked Examples

Example 1

medium

A bag has

5

red,

4

blue, and

1

green marble. What is the probability of NOT drawing a red marble?

Answer

P = \dfrac{1}{2}

First step

Total

= 10

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Example 2

medium

A spinner is divided into sections labeled A, B, C, D with probabilities

0.1, 0.3, 0.4, 0.2

. Are these valid probabilities?

Example 3

hard

A spinner has

8

sections,

3

red and

5

blue. After

80

spins, red came up

24

times. Compute both the theoretical and experimental probability of red.

Example 4

challenge

A bag has

n

red and

n

blue marbles. One marble is drawn. Show that

P(\text{red}) = 1/2

for any

n \geq 1

Example 5

easy

A bag contains 3 red, 5 blue, and 2 green marbles. What is the probability of drawing a blue marble?

Example 6

easy

A fair six-sided die is rolled. What is the probability of rolling an even number?

Practice Problems

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

Example 1

easy

A fair die is rolled once. What is the probability of rolling a 4?

Example 2

easy

A coin is flipped once. What is the probability of getting heads?

Example 3

easy

A bag has 3 red and 2 blue marbles. What is the probability of drawing a red marble?

One marble drawn from a bag of 3 red and 2 blue

Example 4

easy

What is the probability of an impossible event?

Example 5

easy

A spinner has 8 equal sections numbered 1 to 8. What is the probability of landing on an even number?

Example 6

easy

A standard 52-card deck is shuffled. What is the probability of drawing a heart?

Example 7

easy

The probability it rains tomorrow is

0.3

. What is the probability it does NOT rain?

Complement rule: Rain and No Rain together sum to 1

Example 8

easy

Probabilities of all outcomes in a sample space must add up to what number?

Example 9

medium

A bag has 4 green, 3 yellow, and 5 white marbles. What is the probability of drawing a marble that is NOT yellow?

Bag of 12 marbles: 3 yellow versus 9 non-yellow

Example 10

medium

A die is rolled. What is the probability of rolling a number greater than 4?

Example 11

medium

A spinner has sections colored red (probability

0.25

), blue (

0.35

), and green. What is the probability of green?

Example 12

medium

A bag has marbles numbered 1 to 20. What is the probability of drawing a multiple of 5?

Example 13

medium

Two events cover a sample space:

P(A)=2x

and

P(B)=3x

with no other outcomes. Find

x

Example 14

medium

A jar has

6

red marbles,

6

blue marbles, and some green marbles. If

P(\text{green}) = \dfrac{1}{4}

, how many green marbles are there?

Example 15

medium

A die is rolled. What is the probability of rolling a prime number?

Example 16

medium

A fair coin is flipped and the probability of heads is recorded. If heads came up the last 5 flips, what is the probability of heads on the next flip?

Example 17

medium

A card is drawn from a 52-card deck. What is the probability of drawing a red face card (J, Q, K of hearts or diamonds)?

Example 18

challenge

A spinner is split so

P(\text{red})=\frac{1}{2}

P(\text{blue})=\frac{1}{3}

, and the rest is green. What is

P(\text{green})

Example 19

challenge

In a class,

P(\text{plays sport})=0.6

and

P(\text{plays music})=0.5

, and

P(\text{both})=0.2

. What fraction play neither?

Inclusion-exclusion: find the probability of playing neither sport nor music

Example 20

challenge

A bag has

n

red and 8 blue marbles. The probability of red is

\frac{1}{3}

. Find

n

Example 21

easy

A fair die is rolled. What is the probability of rolling a

6

Example 22

easy

A spinner has

10

equal sections numbered

1

10

. What is the probability of landing on

7

Example 23

easy

A bag contains

4

red,

3

blue, and

5

green marbles. What is the probability of drawing a blue marble?

Example 24

easy

A coin is flipped. What is the probability of tails?

Example 25

easy

From a standard deck of

52

cards, what is the probability of drawing a king?

Example 26

medium

A die is rolled. What is the probability of rolling an odd number?

Example 27

medium

A spinner has

12

equal sections numbered

1

12

. What is the probability of landing on a multiple of

3

Example 28

medium

A spinner has red, blue, and green sections with

P(\text{red}) = 0.2

and

P(\text{blue}) = 0.5

. Find

P(\text{green})

Example 29

medium

A bag has tickets numbered

1

30

. What is the probability of drawing a multiple of

7

Example 30

medium

From a deck of

52

cards, find

P(\text{face card})

where face cards are J, Q, K.

Example 31

medium

A bag has

6

yellow and

9

purple marbles. What is the probability of drawing yellow?

Example 32

medium

From

1

50

, find the probability that a randomly chosen number is a perfect square.

Example 33

hard

A die is rolled twice. What is the probability that the first roll is a

6

Example 34

hard

A bag has

20

marbles:

5

red,

8

blue,

7

green. What is the probability of drawing a marble that is red OR blue?

Example 35

hard

A bag has

4

red,

6

green, and

10

blue marbles. Find

P(\text{not green})

Example 36

medium

From a deck of

52

cards, what is the probability of drawing a card that is either red or a queen?

Example 37

hard

A spinner has

5

sections labeled

1, 2, 3, 4, 5

. The probability of landing on each section is proportional to its label. What is the probability of landing on

3

Example 38

easy

A spinner has 8 equal sections numbered 1–8. What is the probability of landing on a number greater than 5?

Example 39

easy

A card numbered 1 through 10 is chosen at random. What is the probability that the number is prime?

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Related Concepts

Relative Frequency Expected Value

Background Knowledge

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

relative frequency