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LearnProbability

Bags, cards and spinners

Find a probability by counting: the outcomes you want divided by all the outcomes, with marbles, cards, letters and spinners.

Most probability questions at school are really counting questions. A bag of marbles, a deck of cards, a spinner, the letters of a word or a number picked from a list: each time, you count what you want, count everything, and make a fraction. The counting is the hard part, and this page shows how to do it carefully, so that you get the right fraction every time.

The counting rule

An outcome is one possible result, like one marble or one card. The outcomes you want are called favorable outcomes. If every outcome is equally likely, the probability is the number of favorable outcomes divided by the number of all outcomes.

Remember

Probability = (favorable outcomes) ÷ (all outcomes)

The bottom of the fraction is ALL the outcomes, including the ones you want.

  1. Count all the outcomes. This is the bottom of the fraction.
  2. Count the outcomes you want. This is the top.
  3. Write the fraction.
  4. Simplify it: divide the top and the bottom by the same number.

Marbles in a bag

You take one marble "without looking", so every marble has the same chance. Add up the marbles of every color to get the bottom number.

Example 1

A bag has 4 red, 3 blue and 5 green marbles. You take one without looking. What is the probability that it is red?

  1. All the marbles: 4 + 3 + 5 = 12.
  2. Red marbles: 4.
  3. Probability: 4/12.
  4. Divide the top and the bottom by 4: 4/12 = 1/3.

Answer: 1/3

Spinners

A spinner is a circle cut into parts, with an arrow that spins and stops at random. If the parts are all the same size, each part is one equally likely outcome. So you count parts, not colors.

Example 2

This spinner has 6 equal parts: 3 yellow, 2 blue and 1 purple. What is the probability that the arrow stops on blue?

  1. All the parts: 6.
  2. Blue parts: 2.
  3. 2/6 = 1/3.

Answer: 1/3

A deck of cards

A normal deck has 52 cards in 4 suits: hearts and diamonds (red), clubs and spades (black). Each suit has 13 cards: an ace, the numbers 2 to 10, and three picture cards, the jack, the queen and the king. So there are 4 aces, 4 kings, 4 of every number, and 13 of every suit. The bottom of the fraction is 52.

You wantHow many cardsProbability
One exact card, like the 7 of clubs11/52
A red king22/52 = 1/26
An ace44/52 = 1/13
A king or a queen4 + 4 = 88/52 = 2/13
A jack, a queen or a king3 × 4 = 1212/52 = 3/13
A heart1313/52 = 1/4
A red card13 + 13 = 2626/52 = 1/2
A number card from 2 to 109 × 4 = 3636/52 = 9/13

Example 3

You take one card from a deck of 52. What is the probability that it is a black picture card (a black jack, queen or king)?

  1. There are 2 black suits, and each has 3 picture cards.
  2. Cards that work: 2 × 3 = 6.
  3. 6/52. Divide the top and the bottom by 2: 3/26.

Answer: 3/26

Letters of a word

Each letter of a word is written on its own card, and you take one card. Every letter counts, also the ones that repeat. The bottom of the fraction is the number of letters in the word.

Example 4

Each letter of the word BALLOON is on its own card. You take one card without looking. What is the probability that it shows an O?

  1. Write the letters: B, A, L, L, O, O, N. That is 7 cards.
  2. O appears 2 times.
  3. Probability: 2/7.

Answer: 2/7

A number chosen at random

When a number from 1 to 20 is chosen "at random", each of the 20 numbers has the same chance. Write the numbers that work, count them, and divide by how many numbers there are. A multiple of 3 is a number in the 3 times table: 3, 6, 9, 12 and so on.

Example 5

A number from 1 to 12 is chosen at random. What is the probability that it is a multiple of 5?

  1. The numbers that work: 5 and 10. That is 2 numbers.
  2. There are 12 numbers in all.
  3. 2/12 = 1/6.

Answer: 1/6

Two-digit numbers are the numbers from 10 to 99. There are 99 − 10 + 1 = 90 of them, not 99 and not 100. So the bottom of the fraction is 90.

Example 6

A two-digit number (10 to 99) is chosen at random. What is the probability that it is a multiple of 10?

  1. The numbers that work: 10, 20, 30, 40, 50, 60, 70, 80, 90. That is 9 numbers.
  2. There are 90 two-digit numbers.
  3. 9/90 = 1/10.

Answer: 1/10

Try it yourself

  1. 1.A bag has 2 red, 6 blue and 4 yellow marbles. You take one without looking. What is the probability that it is yellow?

    Show the answer

    1/3

    There are 2 + 6 + 4 = 12 marbles, and 4 are yellow: 4/12 = 1/3.

  2. 2.A spinner has 8 equal parts: 3 green, 3 red and 2 blue. What is the probability that the arrow stops on green?

    Show the answer

    3/8

    3 of the 8 equal parts are green.

  3. 3.Each letter of the word LETTER is on its own card. You take one. What is the probability that it shows a T?

    Show the answer

    1/3

    LETTER has 6 letters, and 2 of them are T: 2/6 = 1/3.

  4. 4.You take one card from a deck of 52. What is the probability that it is a 2 or a 3?

    Show the answer

    2/13

    There are 4 twos and 4 threes, so 8 cards: 8/52 = 2/13.

  5. 5.A number from 1 to 50 is chosen at random. What is the probability that it is a multiple of 6?

    Show the answer

    4/25

    The multiples are 6, 12, 18, 24, 30, 36, 42 and 48, so 8 numbers: 8/50 = 4/25.

  6. 6.A two-digit number (10 to 99) is chosen at random. What is the probability that it is less than 20?

    Show the answer

    1/9

    The numbers 10 to 19 work. That is 10 of the 90 two-digit numbers: 10/90 = 1/9.

Ready to test yourself?

Play Probability: 20 seconds per question, and it gets harder as you go.