Question
Two fair dice are rolled. What is the probability of
A) rolling a total of 8?
B) rolling a total greater than 5?
C) rolling a 2, 4 times in a row?
D) rolling doubles, or a number that is evenly divisible by 3?
A) rolling a total of 8?
B) rolling a total greater than 5?
C) rolling a 2, 4 times in a row?
D) rolling doubles, or a number that is evenly divisible by 3?
Answers
GPT 3.5
To solve these probabilities, we need to consider all the possible outcomes when rolling two dice.
A) To roll a total of 8, we need the outcome to be {(2,6), (3,5), (4,4), (5,3), (6,2)}. There are 5 favorable outcomes out of a total of 36 possible outcomes when rolling two dice. Therefore, the probability of rolling a total of 8 is 5/36.
B) To roll a total greater than 5, we need the outcome to be {(2,4), (2,5), (2,6), (3,3), (3,4), (3,5), (3,6), (4,2), (4,3), (4,4), (4,5), (4,6), (5,2), (5,3), (5,4), (5,5), (5,6), (6,2), (6,3), (6,4), (6,5), (6,6)}. There are 21 favorable outcomes out of a total of 36 possible outcomes when rolling two dice. Therefore, the probability of rolling a total greater than 5 is 21/36, which can be simplified to 7/12.
C) To roll a 2, 4 times in a row, we need each dice roll to be a 1. The probability of rolling a 1 on one die is 1/6. Since the dice rolls are independent events, we multiply the probabilities together: (1/6) * (1/6) * (1/6) * (1/6) = 1/1296.
D) To roll either doubles or a number evenly divisible by 3, we need the outcome to be {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6), (1,3), (1,6), (2,3), (2,6), (3,1), (3,4), (4,1), (4,3), (5,3), (5,6), (6,1), (6,2)}. There are 18 favorable outcomes out of a total of 36 possible outcomes when rolling two dice. Therefore, the probability of rolling doubles or a number evenly divisible by 3 is 18/36, which can be simplified to 1/2.
A) To roll a total of 8, we need the outcome to be {(2,6), (3,5), (4,4), (5,3), (6,2)}. There are 5 favorable outcomes out of a total of 36 possible outcomes when rolling two dice. Therefore, the probability of rolling a total of 8 is 5/36.
B) To roll a total greater than 5, we need the outcome to be {(2,4), (2,5), (2,6), (3,3), (3,4), (3,5), (3,6), (4,2), (4,3), (4,4), (4,5), (4,6), (5,2), (5,3), (5,4), (5,5), (5,6), (6,2), (6,3), (6,4), (6,5), (6,6)}. There are 21 favorable outcomes out of a total of 36 possible outcomes when rolling two dice. Therefore, the probability of rolling a total greater than 5 is 21/36, which can be simplified to 7/12.
C) To roll a 2, 4 times in a row, we need each dice roll to be a 1. The probability of rolling a 1 on one die is 1/6. Since the dice rolls are independent events, we multiply the probabilities together: (1/6) * (1/6) * (1/6) * (1/6) = 1/1296.
D) To roll either doubles or a number evenly divisible by 3, we need the outcome to be {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6), (1,3), (1,6), (2,3), (2,6), (3,1), (3,4), (4,1), (4,3), (5,3), (5,6), (6,1), (6,2)}. There are 18 favorable outcomes out of a total of 36 possible outcomes when rolling two dice. Therefore, the probability of rolling doubles or a number evenly divisible by 3 is 18/36, which can be simplified to 1/2.