Suppose a pair of dice are rolled. Consider the sum of the numbers on the top of the dice and find the probabilities. (Enter the probabilities as fractions.)
(a) 10 , given that a double was rolled
(b) a double, given that an 10 was rolled
Answer
Final Answer: The probability that the sum of the numbers on the dice is 10, given that a double was rolled is \(\boxed{\frac{1}{6}}\). The probability that a double was rolled, given that the sum of the numbers on the dice is 10 is \(\boxed{\frac{1}{3}}\).
Step 1 ：Suppose a pair of dice are rolled. We are to find the probabilities of the sum of the numbers on the top of the dice.
Step 2 ：For part (a), we need to find the probability that the sum of the numbers on the dice is 10, given that a double was rolled. A double means that the numbers on both dice are the same. The only double that sums to 10 is a pair of 5s. There are 6 possible doubles (1-1, 2-2, 3-3, 4-4, 5-5, 6-6), so the probability is \(\frac{1}{6}\).
Step 3 ：For part (b), we need to find the probability that a double was rolled, given that the sum of the numbers on the dice is 10. The doubles that sum to 10 are 5-5. There are three possible pairs that sum to 10 (4-6, 5-5, 6-4), so the probability is \(\frac{1}{3}\).
Step 4 ：Final Answer: The probability that the sum of the numbers on the dice is 10, given that a double was rolled is \(\boxed{\frac{1}{6}}\). The probability that a double was rolled, given that the sum of the numbers on the dice is 10 is \(\boxed{\frac{1}{3}}\).
