Two dice are rolled and the sum is recorded.(a) List all possible outcomes (i.e. the sample space).(b) Calculate the probability of rolling:(i) A sum of 5(ii) A sum of 4 or less(iii) A sum of at most 11(iv) A sum that is a multiple of 3 or a multiple of 4
Question
Two dice are rolled and the sum is recorded.(a) List all possible outcomes (i.e. the sample space).(b) Calculate the probability of rolling:(i) A sum of 5(ii) A sum of 4 or less(iii) A sum of at most 11(iv) A sum that is a multiple of 3 or a multiple of 4
Solution
(a) The possible outcomes when two dice are rolled can be represented as ordered pairs (x, y), where x and y are numbers from 1 to 6. The sample space is as follows:
{(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}
(b) The total number of outcomes is 36 (6 outcomes for the first die times 6 outcomes for the second die).
(i) The outcomes that give a sum of 5 are {(1,4), (2,3), (3,2), (4,1), (2,3), (3,2), (4,1), (1,4)}. There are 4 such outcomes, so the probability is 4/36 = 1/9.
(ii) The outcomes that give a sum of 4 or less are {(1,1), (1,2), (1,3), (2,1), (2,2), (3,1)}. There are 6 such outcomes, so the probability is 6/36 = 1/6.
(iii) All outcomes give a sum of at most 11, so the probability is 36/36 = 1.
(iv) The outcomes that give a sum that is a multiple of 3 are {(1,2), (1,5), (2,1), (2,4), (3,3), (3,6), (4,2), (4,5), (5,1), (5,4), (6,3)}. There are 11 such outcomes. The outcomes that give a sum that is a multiple of 4 are {(1,3), (2,2), (3,1), (2,6), (3,5), (4,4), (5,3), (6,2)}. There are 8 such outcomes. However, we have double-counted the outcomes that are multiples of both 3 and 4, so we need to subtract those. The outcomes that are multiples of both 3 and 4 are {(2,2), (3,1)}. There are 2 such outcomes. So the total number of outcomes that give a sum that is a multiple of 3 or a multiple of 4 is 11 + 8 - 2 = 17. So the probability is 17/36.
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