Throwing dice: one die and two dice
One die has 6 outcomes. Two dice have 6 × 6 = 36 ordered pairs, like (2, 5). Use the 36-outcome grid to count sums and doublets.
One die
A die (plural dice) is a cube with faces numbered 1 to 6. A fair die shows each face with the same chance, so P(any one number) = 1/6. The prime numbers on a die are 2, 3 and 5. The even numbers are 2, 4 and 6. With two dice, an outcome is a pair: (2, 5) means 2 on the first die and 5 on the second.
Two dice: outcomes for each sum
| Sum | Outcomes (first die, second die) | Count | Probability |
|---|---|---|---|
| 2 | (1, 1) | 1 | 1/36 |
| 3 | (1, 2), (2, 1) | 2 | 2/36 = 1/18 |
| 4 | (1, 3), (2, 2), (3, 1) | 3 | 3/36 = 1/12 |
| 5 | (1, 4), (2, 3), (3, 2), (4, 1) | 4 | 4/36 = 1/9 |
| 6 | (1, 5), (2, 4), (3, 3), (4, 2), (5, 1) | 5 | 5/36 |
| 7 | (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1) | 6 | 6/36 = 1/6 |
| 8 | (2, 6), (3, 5), (4, 4), (5, 3), (6, 2) | 5 | 5/36 |
| 9 | (3, 6), (4, 5), (5, 4), (6, 3) | 4 | 4/36 = 1/9 |
| 10 | (4, 6), (5, 5), (6, 4) | 3 | 3/36 = 1/12 |
| 11 | (5, 6), (6, 5) | 2 | 2/36 = 1/18 |
| 12 | (6, 6) | 1 | 1/36 |
Worked example 1: one die
A die is thrown once. Find the probability of getting (i) a prime number, (ii) a number less than 3.
- Formula: P(E) = favourable outcomes ÷ total outcomes. Total = 6.
- Primes: 2, 3, 5 (3 outcomes). P(prime) = 3/6 = 1/2.
- Less than 3: 1, 2 (2 outcomes). P = 2/6 = 1/3.
P(prime) = 1/2; P(less than 3) = 1/3.
Worked example 2: two dice in a board game
In a board game, two dice are thrown together. Find the probability that (i) the sum is 9, (ii) a doublet appears (the same number on both dice).
- Formula: P(E) = favourable outcomes ÷ total outcomes.
- Total outcomes = 6 × 6 = 36.
- Sum 9: (3, 6), (4, 5), (5, 4), (6, 3). That is 4 outcomes, so P(sum 9) = 4/36 = 1/9.
- Doublets: (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6). That is 6 outcomes.
- P(doublet) = 6/36 = 1/6.
P(sum 9) = 1/9; P(doublet) = 1/6.
How to draw the 36-outcome grid
- Draw a 6 × 6 table.
- Label the rows 1 to 6 for the first die and the columns 1 to 6 for the second die.
- In each cell, write the pair, for example (2, 5), or its sum, 7.
- The cells on the diagonal from top left to bottom right are the doublets.
- Each slanting line from top right to bottom left has the same sum. Count its cells to get the favourable outcomes.
Games like Ludo and Snakes and Ladders depend on a fair die. With two dice, a sum of 7 comes up most often, so players can expect it more than 2 or 12.
1 marks: “A die is thrown once. Find the probability of getting a number that is not a multiple of 3.” Multiples of 3 are 3 and 6. The others are 1, 2, 4, 5 (4). P = 4/6 = 2/3.
3 marks: “Two dice are thrown together. Find the probability that (i) the sum is 8, (ii) the product is 6, (iii) the sum is more than 10.” 36 outcomes. (i) Sum 8: (2, 6), (3, 5), (4, 4), (5, 3), (6, 2), so 5/36. (ii) Product 6: (1, 6), (6, 1), (2, 3), (3, 2), so 4/36 = 1/9. (iii) Sum 11 or 12: (5, 6), (6, 5), (6, 6), so 3/36 = 1/12.
Two dice: 36 pairs. Sum 7 is the peak with 6 ways; doublets also have 6 ways.