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June 2021 p53 q4
3097
Three fair six-sided dice, each with faces marked 1, 2, 3, 4, 5, 6, are thrown at the same time, repeatedly. For a single throw of the three dice, the score is the sum of the numbers on the top faces.
(a) Find the probability that the score is 4 on a single throw of the three dice.
(b) Find the probability that a score of 18 is obtained for the first time on the 5th throw of the three dice.
Solution
(a) To find the probability that the score is 4, we need to consider the possible combinations of dice rolls that sum to 4. The possible combinations are (1, 1, 2), (1, 2, 1), and (2, 1, 1). Each die has a probability of \(\frac{1}{6}\) for each face. Therefore, the probability for each combination is \(\left( \frac{1}{6} \right)^3\). Since there are 3 such combinations, the total probability is:
(b) The probability of obtaining a score of 18 in a single throw is \(\left( \frac{1}{6} \right)^3 = \frac{1}{216}\) because all dice must show a 6. To find the probability that this occurs for the first time on the 5th throw, we use the geometric distribution formula:
