Join the waiting list

IGCSE Maths · Lesson · Unit 8: Probability

How do you use a sample space diagram for two dice?

Draw a two-way table with the first dice down the side and the second along the top, and write the total in each of its 36 cells. Then count: 7 is in 6 cells, so P(7) = 6/36 = 1/6. Every cell is equally likely, but the totals are not.

From the lesson Sample Space, in Probability Diagrams. By the ConceptOrbit team · Updated

The lesson: Sample Space

Sample-space diagrams

In this section

  • List possible outcomes.
  • Use sample-space diagrams for combined events.
  • Calculate probabilities from sample spaces.

A sample space is the list of all possible outcomes, and writing it out turns a probability question into counting.

For two dice, a two-way table with the first die down the side and the second along the top gives all 36 outcomes. Each cell can hold the total, and then counting the cells with a given total answers the question directly: seven appears six times, so P(7) = 6/36 = 1/6.

The table also makes visible what a list does not: the outcomes are equally likely as cells, but the totals are not equally likely as totals. That is why 7 is more likely than 2, and why guessing from the list of possible totals gives the wrong answer.

For two coins the sample space is HH, HT, TH, TT — four outcomes, not three. Treating HT and TH as one outcome is a classic error, and writing them out separately prevents it.

Key points

  • Write out or tabulate every outcome; then it is just counting.
  • For two dice there are 36 equally likely cells, but the totals are not equally likely.
  • HT and TH are different outcomes.

Now try it

Probability with two dice

Probability with two diceMaths · Probability

Roll two dice and add the scores. Which total will come up most often?

Make your prediction first.
5%10%15%20%23456789101112

Your rolls, as a share of all of them

Rolls
0
Most common so far
—
Seven so far
—
Expect 6 in 36, about 17%

Tasks0 of 3 done

  • Roll the dice at least 100 times.
  • Show what to expect, and compare it with 100 or more rolls.
  • Keep rolling until 7 comes up within 2% of 1 in 6 (1000 rolls or more).
Open Probability with two dice on its own page

Common mistakes

  • Treating the 11 totals from 2 to 12 as equally likely.
  • Expecting the relative frequency to equal the probability exactly after many throws.

More lessons, each with its simulation