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Maths · Probability

Combined events

List the outcomes of two events systematically, use sample space diagrams, and work out probabilities of combined events.

  • 6 key terms
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Warm-up

Answer each one, then check.

  1. 1

    What is the probability of rolling a 3 on a fair dice?

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    \(\dfrac{1}{6}\)

  2. 2

    What is the probability of getting tails on a fair coin?

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    \(\dfrac{1}{2}\)

  3. 3

    Simplify \(\dfrac{6}{36}\).

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    \(\dfrac{1}{6}\)

  4. 4

    What do all the probabilities of an event add up to?

    Show answerHide answer

    1

  5. 5

    If P(rain) = 0.3, what is P(not rain)?

    Show answerHide answer

    0.7

Learning Objectives

  1. 1Use the probability scale and the formula for equally likely outcomes.
  2. 2List the outcomes of two events systematically.
  3. 3Draw and use a sample space diagram.
  4. 4Work out probabilities of combined events.

THE FORMULA

For equally likely outcomes, probability is the number of successful outcomes divided by the total number of possible outcomes.

\(P(\text{event}) = \dfrac{\text{number of ways the event can happen}}{\text{total number of equally likely outcomes}}\)

Listing Outcomes

A fair coin is flipped and a fair dice is rolled. List all the possible outcomes, and find the probability of a head and a 6.

Show the solutionHide the solution
  1. 1 Write each coin outcome with each dice outcome H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6
  2. 2 Count the outcomes 12
  3. 3 Only one of these is a head with a 6 H6

AnswerThere are 12 outcomes, so \(P(\text{head and } 6) = \dfrac{1}{12}\).

Totals from Two Dice

Each cell is the sum of the two dice.

  • 1

    1: 2. 2: 3. 3: 4. 4: 5 | 6 | 7

  • 2

    1: 3. 2: 4. 3: 5. 4: 6 | 7 | 8

  • 3

    1: 4. 2: 5. 3: 6. 4: 7 | 8 | 9

  • 4

    1: 5. 2: 6. 3: 7. 4: 8 | 9 | 10

  • 5

    1: 6. 2: 7. 3: 8. 4: 9 | 10 | 11

  • 6

    1: 7. 2: 8. 3: 9. 4: 10 | 11 | 12

Using the Table

Two fair dice are rolled and the scores are added. Find (a) \(P(\text{total is } 7)\) and (b) \(P(\text{total is at least } 10)\).

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  1. 1 (a) Count the 7s in the table 6 out of 36
  2. 2 (a) Simplify \(\dfrac{6}{36} = \dfrac{1}{6}\)
  3. 3 (b) Count the totals 10, 11 or 12 \(3 + 2 + 1 = 6\)
  4. 4 (b) Probability \(\dfrac{6}{36} = \dfrac{1}{6}\)

Answer(a) \(\dfrac{1}{6}\) (b) \(\dfrac{1}{6}\)

Two Spinners

Spinner A has three equal sections numbered 1, 2, 3. Spinner B has four equal sections numbered 1, 2, 3, 4. Both are spun and the scores added. Find the probability that the total is 5, and that it is even.

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  1. 1 Number of outcomes \(3 \times 4 = 12\)
  2. 2 Totals of 5 come from (1, 4), (2, 3), (3, 2)
  3. 3 \(P(\text{total is } 5)\) \(\dfrac{3}{12} = \dfrac{1}{4}\)
  4. 4 Even totals: both odd or both even (1,1), (1,3), (3,1), (3,3), (2,2), (2,4)
  5. 5 \(P(\text{even})\) \(\dfrac{6}{12} = \dfrac{1}{2}\)

Answer\(P(\text{total } 5) = \dfrac{1}{4}\) and \(P(\text{even}) = \dfrac{1}{2}\).

Systematic Listing

A system stops you missing outcomes or counting one twice.

  • Fix the first event

    Write every second-event outcome beside it, then move to the next first-event outcome.

  • Use a table

    Rows for one event, columns for the other.

  • Count

    The total number of outcomes is the product of the number of outcomes of each event.

Two-Dice Game

In a game, two dice are rolled and the smaller number is subtracted from the larger (or 0 if they are equal). Complete a table of differences. Which difference is most likely, and what is its probability?

1. Fill in a 6 by 6 table.

2. Count each difference.

3. Divide by 36.

A good answer shows: The differences 0 to 5 occur 6, 10, 8, 6, 4 and 2 times out of 36. The most likely is 1, with probability \(\dfrac{10}{36} = \dfrac{5}{18}\).

Can I...?

  1. 1Write a probability as a fraction.
  2. 2List outcomes systematically.
  3. 3Draw a sample space diagram.
  4. 4Count successful outcomes.
  5. 5Work out probabilities from a table.
  6. 6Simplify probabilities.
  7. 7Use \(3 \times 4 = 12\) to count outcomes.
  8. 8Avoid double-counting.

Summary & Exam Focus

  • Probability \(= \dfrac{\text{successful outcomes}}{\text{total outcomes}}\).
  • Use a table or systematic list to find all outcomes.
  • The total number of outcomes for two events is the product.
  • Simplify your final fraction.

Exam focus

Two fair dice are rolled and their scores are added. Work out the probability that the total is 7. (3 marks) (3 marks)

Draw a 6 by 6 table. Count the 7s (there are 6) and divide by 36.

Key terms

The vocabulary this lesson expects you to use. Each one is linked from the first place it appears above.

Outcome
One possible result of an event.
Event
One or more outcomes.
Equally likely
Each outcome has the same chance.
Sample space
The set of all possible outcomes.
Sample space diagram
A table or list showing every outcome.
Probability
A number from 0 to 1 showing how likely an event is.

Practice questions

Have a go at each one before you open its answer.

  1. Question 1 Non-calculator 2 marks

    A fair coin is flipped and a fair dice is rolled. Write down the probability of getting a head and a 6.

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    Model answer

    There are 12 equally likely outcomes and one of them is H6. The probability is \(\dfrac{1}{12}\).

    Mark scheme

    • 12 outcomes — M1
    • \(\dfrac{1}{12}\) — A1
  2. Question 2 Non-calculator 3 marks

    Two fair dice are rolled and the scores are added. Work out the probability that the total is 7.

    Show answerHide answer

    Model answer

    There are 36 equally likely outcomes and 6 give a total of 7. The probability is \(\dfrac{6}{36} = \dfrac{1}{6}\).

    Mark scheme

    • 36 outcomes — M1
    • 6 successful outcomes — M1
    • \(\dfrac{1}{6}\) — A1
  3. Question 3 Non-calculator 3 marks

    Spinner A has three equal sections numbered 1, 2 and 3. Spinner B has four equal sections numbered 1, 2, 3 and 4. Both spinners are spun and the two scores are added. Work out the probability that the total is 5.

    Spinner A with three equal sections numbered 1, 2, 3 and spinner B with four equal sections numbered 1, 2, 3, 4.
    Show answerHide answer

    Model answer

    There are \(3 \times 4 = 12\) outcomes. The totals of 5 are (1, 4), (2, 3) and (3, 2), so the probability is \(\dfrac{3}{12} = \dfrac{1}{4}\).

    Mark scheme

    • 12 outcomes — M1
    • Three ways to make 5 — M1
    • \(\dfrac{1}{4}\) — A1
  4. Question 4 Non-calculator 2 marks

    For the same two spinners, work out the probability that the total is even.

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    Model answer

    The even totals come from (1,1), (1,3), (3,1), (3,3), (2,2) and (2,4): 6 of 12. The probability is \(\dfrac{1}{2}\).

    Mark scheme

    • 6 successful outcomes — M1
    • \(\dfrac{1}{2}\) — A1
  5. Question 5 Non-calculator 3 marks

    Two fair dice are rolled and the scores are added. Work out the probability that the total is at least 10.

    Show answerHide answer

    Model answer

    Totals of 10, 11 and 12 occur \(3 + 2 + 1 = 6\) times out of 36. The probability is \(\dfrac{6}{36} = \dfrac{1}{6}\).

    Mark scheme

    • 6 successful outcomes — M1
    • \(\dfrac{6}{36}\) — M1
    • \(\dfrac{1}{6}\) — A1
  6. Question 6 Non-calculator 2 marks

    Two fair dice are rolled. Work out the probability of getting a double (the same number on both dice).

    Show answerHide answer

    Model answer

    There are 6 doubles out of 36 outcomes, so the probability is \(\dfrac{6}{36} = \dfrac{1}{6}\).

    Mark scheme

    • 6 doubles — M1
    • \(\dfrac{1}{6}\) — A1

Quick check

  1. A fair dice is rolled. What is the probability of an even number?

    1. A\(\dfrac{1}{6}\)
    2. B\(\dfrac{1}{2}\)
    3. C\(\dfrac{1}{3}\)
    4. D\(\dfrac{2}{3}\)
    Show answerHide answer

    B: \(\dfrac{1}{2}\)

    3 even numbers out of 6: \(\dfrac{3}{6} = \dfrac{1}{2}\).

  2. How many outcomes are there when a coin is flipped and a dice is rolled?

    1. A8
    2. B6
    3. C12
    4. D36
    Show answerHide answer

    C: 12

    \(2 \times 6 = 12\).

  3. How many outcomes are there when two dice are rolled?

    1. A12
    2. B6
    3. C24
    4. D36
    Show answerHide answer

    D: 36

    \(6 \times 6 = 36\).

  4. Two fair dice are rolled. What is the probability of a total of 2?

    1. A\(\dfrac{1}{36}\)
    2. B\(\dfrac{1}{12}\)
    3. C\(\dfrac{1}{6}\)
    4. D\(\dfrac{2}{36}\)
    Show answerHide answer

    A: \(\dfrac{1}{36}\)

    Only (1, 1): \(\dfrac{1}{36}\).

  5. What is the best way to make sure no outcome is missed?

    1. AGuess
    2. BList them systematically
    3. COnly count the first few
    4. DRoll the dice
    Show answerHide answer

    B: List them systematically

    Use a systematic list or a sample space table.

  6. Two fair dice are rolled. Which total is most likely?

    1. A6
    2. B8
    3. C7
    4. D12
    Show answerHide answer

    C: 7

    7 can be made 6 ways, more than any other total.

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