Probability Examples in Math

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

Probability is a number between 0 and 1 (inclusive) that measures how likely an event is to occur, where 0 means impossible and 1 means certain.

How confident you should be that something will happen. 0 = impossible, 1 = certain.

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: Probability counts the ways something can happen against all the equally-likely ways anything can happen.

Common stuck point: The procedure for probability is the easy part; the trap is counting outcomes that are not equally likely. Asking "Are the outcomes equally likely, and am I asked how likely (not how many)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Are the outcomes equally likely, and am I asked how likely (not how many)?

Worked Examples

Example 1

easy
A bag contains 55 red, 33 blue, and 22 green marbles. What is the probability of drawing a blue marble?

Answer

P(blue)=310P(\text{blue}) = \frac{3}{10}

First step

1
Total number of marbles: 5+3+2=105 + 3 + 2 = 10.

Full solution

  1. 2
    Number of favorable outcomes (blue): 33.
  2. 3
    Probability: P(blue)=310=0.3P(\text{blue}) = \frac{3}{10} = 0.3.
Basic probability is the ratio of favorable outcomes to total outcomes. Probability values always fall between 00 (impossible) and 11 (certain).

Example 2

medium
Two fair dice are rolled. What is the probability that the sum is 77?

Example 3

medium
A bag contains 4 red, 3 blue, and 5 green marbles. What is the probability of drawing a red or blue marble?

Example 4

medium
Two coins are tossed. What is the probability of exactly one head?

Example 5

medium
A card is drawn from a deck. P(red queen)?

Example 6

medium
A die is rolled twice. P(at least one 66)?

Example 7

medium
P(sum of two dice is a multiple of 4)?

Practice Problems

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

Example 1

medium
A card is drawn at random from a standard 5252-card deck. What is the probability it is a heart or a king?

Example 2

medium
A box contains 44 red, 55 blue, and 11 green marble. Two marbles are drawn without replacement. What is the probability of getting one red and one blue in any order?

Example 3

easy
A fair die is rolled. What is the probability of rolling a 44?

Example 4

easy
A coin is flipped. What is the probability of heads?

Example 5

easy
A bag has 33 red and 55 blue marbles. P(red)?

Example 6

easy
What is the probability of an impossible event?

Example 7

easy
A die is rolled. P(even number)?

Example 8

easy
P(not rolling a 6) on a fair die?

Example 9

easy
A spinner has 44 equal sections: red, blue, green, yellow. P(blue)?

Example 10

easy
A die is rolled. P(rolling a 7)?

Example 11

medium
Two coins are flipped. P(two heads)?

Example 12

medium
A die is rolled. P(rolling a 2 or a 5)?

Example 13

medium
A bag has 44 red, 66 blue. Draw two WITHOUT replacement. P(both red)?

Example 14

medium
Roll two dice. P(sum of 7)?

Example 15

medium
A spinner lands on red with P = 0.30.3. What's P(not red)?

Example 16

medium
Flip a coin 33 times. P(at least one head)?

Example 17

medium
A jar has 22 red, 33 green, 55 blue. P(red or green)?

Example 18

medium
Roll a die twice. P(both even)?

Example 19

medium
If P(rain) = 25\frac{2}{5}, what are the odds in favor of rain?

Example 20

challenge
Three coins are flipped. P(exactly two heads)?

Example 21

challenge
A bag has 55 red and 55 blue. Draw 22 without replacement. P(one of each color)?

Example 22

challenge
The gambler's fallacy: after 55 heads in a row, is tails 'more likely' on the next flip? Explain.

Example 23

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

Example 24

easy
A bag has 22 red, 33 blue, and 55 green marbles. P(green)?

Example 25

medium
Two fair dice are rolled. P(sum =8=8)?

Example 26

medium
A card is drawn from a standard deck. P(face card: J/Q/K)?

Example 27

medium
Two fair dice are rolled. P(both show 66)?

Example 28

hard
A bag has 44 red and 66 blue marbles. Two are drawn without replacement. P(both red)?

Example 29

easy
A spinner has 5 equal sections numbered 1-5. P(prime)?

Example 30

medium
A coin is flipped 3 times. P(exactly 2 heads)?

Example 31

medium
From a deck, two cards drawn with replacement. P(both aces)?

Example 32

easy
In a class of 2020, 1212 are girls. A student is picked at random. P(girl)?

Example 33

hard
A bag has 33 red and 55 blue marbles. Two drawn without replacement. P(one red, one blue)?

Example 34

medium
In a deck, P(spade or face card)?

Example 35

easy
A coin is flipped. P(tails)?

Example 36

hard
Three coins are tossed. P(at least one tail)?

Example 37

medium
A bag has 77 marbles: 33 red and 44 blue. P(red on a single draw)?

Example 38

hard
Two dice are rolled. P(product is odd)?

Example 39

challenge
In a class of 2525 students, 3 are on the math team. A pair of students is chosen at random. What is the probability that at least one math-team member is in the pair?
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Background Knowledge

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

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