A fair six-sided die is rolled and a fair three-sided spinner with equal-sized sections labelled 1, 4, and 8 is spun.
The results are added together.
What is the probability that the total is greater than 8?
Give your answer as a fraction.
Answer (2)
List all possible outcomes of rolling a six-sided die and spinning a three-sided spinner.
Calculate the sum of each outcome.
Count the number of outcomes where the sum is greater than 8, which is 8.
Divide the number of favorable outcomes by the total number of outcomes to find the probability: 18 8 = 9 4 .
Explanation
Understand the problem and provided data We are given a fair six-sided die and a fair three-sided spinner. The die has outcomes {1, 2, 3, 4, 5, 6}, each with probability 6 1 . The spinner has outcomes {1, 4, 8}, each with probability 3 1 . We want to find the probability that the sum of the die roll and spinner result is greater than 8.
List possible outcomes and calculate sums Let's list all possible outcomes of the die roll and spinner and calculate their sums. The total number of outcomes is 6 × 3 = 18 . We will then identify the outcomes where the sum is greater than 8.
Calculate all possible sums Here are all the possible outcomes and their sums:
(1, 1) = 2 (1, 4) = 5 (1, 8) = 9 (2, 1) = 3 (2, 4) = 6 (2, 8) = 10 (3, 1) = 4 (3, 4) = 7 (3, 8) = 11 (4, 1) = 5 (4, 4) = 8 (4, 8) = 12 (5, 1) = 6 (5, 4) = 9 (5, 8) = 13 (6, 1) = 7 (6, 4) = 10 (6, 8) = 14
Identify sums greater than 8 Now, let's identify the outcomes where the sum is greater than 8:
(1, 8) = 9 (2, 8) = 10 (3, 8) = 11 (4, 8) = 12 (5, 4) = 9 (5, 8) = 13 (6, 4) = 10 (6, 8) = 14
There are 8 outcomes where the sum is greater than 8.
Calculate the probability The probability that the sum is greater than 8 is the number of favorable outcomes (sum > 8) divided by the total number of possible outcomes. In this case, it is 18 8 . We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2. So, 18 8 = 9 4 .
State the final answer Therefore, the probability that the total is greater than 8 is 9 4 .
Examples
This type of probability problem can be used in games of chance. For example, if you were designing a board game where players roll a die and spin a spinner to determine how many spaces they move, you could use this calculation to determine the likelihood of landing on certain spaces based on the total movement points.
The probability that the total of rolling a die and spinning a spinner is greater than 8 is 9 4 . This is determined by identifying all possible outcomes and counting the favorable ones that exceed a sum of 8. The total outcomes are 18, with 8 being favorable for this condition.
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