Chance Examples in Math

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

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

Concept Recap

Chance describes the inherent randomness in outcomes of experiments — the fact that even with complete knowledge, some events cannot be predicted with certainty.

When multiple outcomes are possible and we can't control which occurs.

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: Chance is the everyday idea that some outcomes can't be known for sure ahead of time.

Common stuck point: The procedure for chance is the easy part; the trap is calling a guaranteed event a matter of chance. Asking "Could this go more than one way, with no way to control which?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Could this go more than one way, with no way to control which?

Worked Examples

Example 1

easy
A bag contains 3 red, 2 blue, and 5 green marbles. Find the probability of randomly drawing a red marble and express it as a fraction, decimal, and percentage.

Answer

P(Red)=310=0.30=30%P(\text{Red}) = \frac{3}{10} = 0.30 = 30\%

First step

1
Total marbles: 3+2+5=103 + 2 + 5 = 10

Full solution

  1. 2
    Favorable outcomes: 3 (red marbles)
  2. 3
    Probability: P(Red)=310P(\text{Red}) = \frac{3}{10}
  3. 4
    As decimal: 0.300.30; as percentage: 30%30\%
Probability quantifies chance as the ratio of favorable outcomes to total equally likely outcomes. All three representations (fraction, decimal, percent) are equivalent — use whichever is clearest for the context.

Example 2

medium
Weather forecasts say 70% chance of rain tomorrow. Explain what this probability means in terms of relative frequency, and how it differs from certainty.

Example 3

medium
The weather app says 30%30\% chance of rain. If you check on 100100 similar days, about how many should be rainy?

Example 4

medium
After flipping 55 tails in a row with a fair coin, what is the chance the next flip is heads?

Example 5

medium
Why is 'something will happen' (any outcome at all) an event with probability 11?

Example 6

medium
Why is 'I won the coin toss once, so I'll lose next time' a poor argument?

Example 7

hard
Two fair dice are rolled. What is the chance the sum is 77?

Practice Problems

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

Example 1

easy
A spinner has 8 equal sections numbered 1–8. Find (a) P(even)P(\text{even}) and (b) P(greater than 5)P(\text{greater than 5}).

Example 2

hard
In a game, you roll a die. If you roll 6, you win $5\$5. Otherwise you lose $1\$1. Calculate the expected value per game and determine if the game is fair.

Example 3

easy
A fair die is rolled. What is the chance of rolling a 3?

Example 4

easy
Does a 1% chance event mean it will never happen?

Example 5

easy
Is a 50% chance the same as 'any chance at all'?

Example 6

easy
Two outcomes are possible and we cannot control which occurs. Is this a situation governed by chance?

Example 7

easy
A coin lands heads. Was the outcome certain before the flip?

Example 8

easy
A bag has 3 red and 7 blue marbles. What is the chance of drawing red?

Example 9

easy
If something has a 0% chance, is it possible?

Example 10

easy
Over a huge population, will a 1-in-a-million daily event likely occur somewhere each day if there are 10 million trials?

Example 11

medium
Two fair coins are tossed. What is the chance of getting exactly one head?

Example 12

medium
A weather app says 70% chance of rain today and it does not rain. Was the forecast wrong?

Example 13

medium
After 20 reds in a row at roulette (red probability ≈18/38\approx 18/38), is black 'due' on the next spin?

Example 14

medium
A 1-in-1000 chance machine error occurs. The operator says 'impossible, we've run it 500 times fine.' Critique this reasoning.

Example 15

medium
A spinner is split: 60% of the area is green, 40% red. What is the chance of green, and why does area matter?

Example 16

medium
Why can the gap between observed heads and expected heads grow even as the proportion approaches 0.5?

Example 17

medium
A fair die is rolled twice. What is the chance of getting a 6 on at least one of the two rolls?

Example 18

medium
A jar has 2 winning tickets among 10. You draw one ticket. The chance of winning is 1/5. If 100 people each draw from identical jars, about how many win?

Example 19

medium
A game has a 1/4 chance of success each independent try. What is the chance of failing all of 3 tries?

Example 20

challenge
A bag has 4 red and 6 blue. Two marbles are drawn without replacement. What is the chance both are red?

Example 21

challenge
A test for a rare disease (1% of people) is 90% accurate both ways. If you test positive, is the chance you have the disease close to 90%? Estimate it.

Example 22

challenge
A '1 in 13 million' lottery is played by 20 million distinct tickets in one draw. Is it likely at least one ticket wins? Estimate.

Example 23

easy
A bag has 44 red, 11 blue, and 55 yellow marbles. What is the chance of drawing a blue marble?

Example 24

easy
A spinner has 1010 equal slices, of which 22 are red. What is the chance of landing on red?

Example 25

easy
A standard deck of 5252 cards is shuffled. What is the chance of drawing a heart?

Example 26

medium
Two coins are flipped. What is the chance both land heads?

Example 27

medium
A spinner has 55 equal slices labeled 11 through 55. What is the chance of an odd number?

Example 28

easy
True or false: 'unlikely' and 'impossible' mean the same thing.

Example 29

medium
A fair six-sided die is rolled. What is the chance the result is greater than 44?

Example 30

medium
A box has 2020 tickets, 33 labeled 'win'. What is the chance of drawing a win on one draw?

Example 31

easy
A bag has only red marbles. What is the chance of drawing a red marble?

Example 32

easy
A bag has only blue marbles. What is the chance of drawing a red marble?

Example 33

medium
A spinner has 44 equal sections: red, blue, green, yellow. What is the chance of NOT landing on red?

Example 34

medium
A jar has 66 red, 44 blue marbles. What is the chance the next draw is red or blue?

Example 35

hard
A bag has 77 blue and 33 red marbles. You draw without looking and replace before drawing again. What is the chance of two blues in a row?

Example 36

hard
A bag has 77 blue and 33 red marbles. You draw without replacement. What is the chance of two blues in a row?

Example 37

hard
If P(A)=0.4P(A) = 0.4 and P(B)=0.5P(B) = 0.5 with AA and BB independent, find P(A and B)P(A \text{ and } B).

Example 38

medium
A fair die is rolled. Compute P(prime)P(\text{prime}). (Primes among 11–66: 2,3,52,3,5.)

Example 39

challenge
A class of 3030 students has a 0.010.01 chance per student per day of being absent. What's the chance at least one student is absent on a given day, assuming independence?
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Related Concepts

ProbabilityRisk