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Academy11+

11+ Maths · Topic explained

Probability of two events

Toss a coin and roll a dice: how many results could you get?

The big idea

For two events, list every pair in a grid, then count the ones you want.

Picture it

Like picking an outfit: 2 tops and 3 pairs of trousers make 2 × 3 = 6 outfits.

Key words

outcome
One possible result, like heads and a 5.
sample space
The full list (or grid) of every possible outcome.
systematic
Listing in a fixed order so nothing is missed or repeated.

Worked examples

  1. 1Warm-up

    Two coins are tossed. P(two heads) = ?

    • Outcomes: HH, HT, TH, TT
    • Only HH is two heads
    • P(two heads) = 1/4
  2. 211+ standard

    Two dice are rolled and the scores added. P(total of 10) = ?

    • 6 × 6 = 36 outcomes
    • 4 + 6, 5 + 5, 6 + 4: 3 ways
    • P = 3/36 = 1/12
  3. 311+ standard

    A coin is tossed and a dice is rolled. P(heads and an even number) = ?

    • 2 × 6 = 12 outcomes
    • H2, H4, H6: 3 of them
    • P = 3/12 = 1/4
  4. 411+ standard

    Spinner A shows 1, 2 or 3. Spinner B shows 1, 2, 3 or 4. The two numbers are multiplied. P(odd answer) = ?

    • 3 × 4 = 12 outcomes
    • Odd × odd only: 1 or 3 with 1 or 3 = 4 ways
    • P(odd) = 4/12 = 1/3

The common mistake

The mistake

Two coins: two heads, two tails or one of each, so P(one of each) = 1/3.

Instead

There are 4 outcomes, because HT and TH are different: P(one of each) = 2/4 = 1/2.

Why: Coin 1 heads and coin 2 tails is a different outcome from coin 1 tails and coin 2 heads.

Where it comes up in the 11+

Dice totals, two spinners, coins with dice: draw the grid or list systematically, then count.

The video

Video coming soon

We're making a short video that explains probability of two events step by step. Until then, everything it covers is on this page.

Free printable

Probability of two events worksheet

Practice questions on this topic to print and try on paper, at home or on the way to school.

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Quick quiz

Have a go first, then tap to see the answer.

  1. 1. Two coins are tossed. What is the probability of two tails?

    • 1/2
    • 1/3
    • 1/4
    • 2/3
    Show the answer

    1/4

    HH, HT, TH, TT: only TT is two tails, 1 out of 4.

  2. 2. A coin is tossed and a dice is rolled. How many outcomes are there?

    • 8
    • 12
    • 6
    • 36
    Show the answer

    12

    2 sides × 6 faces = 12.

  3. 3. Two dice are rolled. What is the probability of a double?

    • 1/6
    • 1/36
    • 1/2
    • 1/12
    Show the answer

    1/6

    6 doubles (1 and 1 up to 6 and 6) out of 36 outcomes: 6/36 = 1/6.