Question: If two fair dice are rolled, find the probability of the following result. A double, given that the…

If two fair dice are rolled, find the probability of the following result. A double, given that the sum was 6. … The probability is \boxed{\rule{0.5in}{0.1pt}}. (Type an integer or a simplified fraction.)

Solution

To find the probability of rolling a double given that the sum is 6, we start by identifying all possible outcomes where the sum of two dice is 6. The pairs that give a sum of 6 are: - (1, 5) - (2, 4) - (3, 3) - (4, 2) - (5, 1) The total number of outcomes where the sum is 6 is 5. Next, identify the favorable outcomes where a double (both dice show the same number) occurs. In this case, the only double is: - (3, 3) There is 1 favorable outcome. Now, calculate the probability: \[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \] \[ \text{Probability} = \frac{1}{5} \] Thus, the probability of rolling a double given that the sum is 6 is \(\frac{1}{5}\).