Answer:
Explanation:
To calculate the probabilities for the various outcomes when rolling a pair of six-sided dice, you can use a probability distribution table or a visual representation of all possible outcomes. There are 36 equally likely outcomes when rolling two dice (6 faces on the first die times 6 faces on the second die).
(a) Probability that the sum of the numbers on your dice is exactly 4:
You can achieve a sum of 4 in the following ways:
1. (1, 3)
2. (2, 2)
3. (3, 1)
There are 3 favorable outcomes.
Probability = (Number of favorable outcomes) / (Total number of possible outcomes) = 3 / 36 = 1 / 12
(b) Probability that the sum of the numbers on your dice is at most 7:
To find the probability of getting a sum at most 7, you need to calculate the probability of getting a sum of 2, 3, 4, 5, 6, or 7. You've already calculated the probability of a sum of 4 in part (a). Now, for the other sums:
Sum of 2:
There's one way to get a sum of 2: (1, 1).
Sum of 3:
There are two ways to get a sum of 3: (1, 2) and (2, 1).
Sum of 5:
There are four ways to get a sum of 5: (1, 4), (2, 3), (3, 2), and (4, 1).
Sum of 6:
There are five ways to get a sum of 6: (1, 5), (2, 4), (3, 3), (4, 2), and (5, 1).
Sum of 7:
There are six ways to get a sum of 7: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1).
Total favorable outcomes for sums at most 7: 1 (sum of 2) + 2 (sum of 3) + 3 (sum of 4) + 4 (sum of 5) + 5 (sum of 6) + 6 (sum of 7) = 21 favorable outcomes.
Probability = (Number of favorable outcomes) / (Total number of possible outcomes) = 21 / 36 = 7 / 12
(c) Probability that the sum of the numbers on your dice is at least 10:
To find the probability of getting a sum at least 10, you need to calculate the probability of getting a sum of 10 or 11 or 12.
Sum of 10:
There are three ways to get a sum of 10: (4, 6), (5, 5), and (6, 4).
Sum of 11:
There are two ways to get a sum of 11: (5, 6) and (6, 5).
Sum of 12:
There's one way to get a sum of 12: (6, 6).
Total favorable outcomes for sums at least 10: 3 (sum of 10) + 2 (sum of 11) + 1 (sum of 12) = 6 favorable outcomes.
Probability = (Number of favorable outcomes) / (Total number of possible outcomes) = 6 / 36 = 1 / 6