IGCSE Maths · Simulation · C8.2, C8.3, E8.2, E8.3 Core and Extended
Probability with two dice
What is the most likely total when you throw two dice?
7 is the most likely total with two fair dice. There are 36 equally likely outcomes, and 6 of them make 7 (1 + 6, 2 + 5, 3 + 4, 4 + 3, 5 + 2 and 6 + 1), so P(7) = 6/36 = 1/6. The totals from 2 to 12 are not equally likely: 2 and 12 can each happen only one way.
Roll two dice and add the scores. Which total will come up most often?
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).
Free to use here, with no account.
What to take away
- When outcomes are equally likely, probability = number of outcomes in the event ÷ total number of outcomes.
- Relative frequency = number of times it happened ÷ number of trials, and it estimates the probability.
- Over more trials the relative frequency usually gets closer to the probability, but it rarely equals it.
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.
Questions
What is the most likely total when you throw two dice?
7 is the most likely total with two fair dice. There are 36 equally likely outcomes, and 6 of them make 7 (1 + 6, 2 + 5, 3 + 4, 4 + 3, 5 + 2 and 6 + 1), so P(7) = 6/36 = 1/6. The totals from 2 to 12 are not equally likely: 2 and 12 can each happen only one way.
How many outcomes are there when you throw two dice?
36. Each of the 6 faces of the first dice can go with each of the 6 faces of the second, and all 36 pairs are equally likely with fair dice.
What is relative frequency?
The number of times something happened divided by the number of trials. It estimates the probability, and usually gets closer to it as the number of trials grows.
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