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?
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 find the probability of rolling a total of 8, we need to count the number of possible outcomes that result in a total of 8 and divide it by the total number of possible outcomes.
The possible outcomes that result in a total of 8 are:
(2, 6), (3, 5), (4, 4), (5, 3), and (6, 2).
There are 5 possible outcomes that result in a total of 8. Since there are 36 total possible outcomes when rolling two dice (each with 6 possible outcomes), the probability is:
P(rolling a total of 8) = 5/36 = 0.1389 (rounded to four decimal places).
B) To find the probability of rolling a total greater than 5, we need to count the number of possible outcomes that result in a total greater than 5 and divide it by the total number of possible outcomes.
The possible outcomes that result in a total greater than 5 are:
(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 possible outcomes that result in a total greater than 5. Therefore, the probability is:
P(rolling a total greater than 5) = 21/36 = 0.5833 (rounded to four decimal places).
C) To find the probability of rolling a 2, 4 times in a row, we need to calculate the probability of getting a 2 on a single roll and raise it to the power of 4 (for 4 consecutive rolls).
The probability of getting a 2 on a single roll is 1/6 since there is one outcome of rolling a 2 out of the six possible outcomes.
Therefore, the probability of rolling a 2, 4 times in a row is:
P(rolling a 2, 4 times in a row) = (1/6)^4 = 1/1296 = 0.0007716 (rounded to four decimal places).
D) To find the probability of rolling either doubles or a number that is evenly divisible by 3, we need to calculate the probability of each event separately and then add the probabilities together.
Probability of rolling doubles:
There are 6 possible outcomes of rolling doubles: (1,1), (2,2), (3,3), (4,4), (5,5), and (6,6). Out of the 36 total possible outcomes, the probability of rolling doubles is 6/36 = 1/6.
Probability of rolling a number that is evenly divisible by 3:
There are 12 possible outcomes that result in a number divisible by 3: (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (6,1), (6,2), (6,3), (6,4), (6,5), and (6,6). So, the probability is 12/36 = 1/3.
Now, we add the probabilities together:
P(rolling doubles or a number divisible by 3) = P(rolling doubles) + P(rolling divisible by 3)
= 1/6 + 1/3 = 1/6 + 2/6 = 3/6 = 1/2 = 0.5