a) A and B are mutually exclusive because if we roll doubles, we cannot roll a sum of 8 and if we roll a sum of 8, we cannot roll doubles.
b) P(A or B) = P(A) + P(B) - P(AB)
To find P(A), we can list all the possible combinations of dice rolls that add up to 8: (2,6), (3,5), (4,4), (5,3), (6,2). The probability of each combination is 1/36, since there are 36 possible outcomes when rolling two dice. Therefore, P(A) = 5/36.
To find P(B), we can list all the possible combinations of doubles: (1,1), (2,2), (3,3), (4,4), (5,5), (6,6). The probability of each combination is 1/36, since there are 36 possible outcomes when rolling two dice. Therefore, P(B) = 6/36 = 1/6.
To find P(AB), we need to find the probability of rolling doubles and getting a sum of 8 at the same time. There are only two possible outcomes that satisfy this condition: (4,4) and (5,3). Therefore, P(AB) = 2/36 = 1/18.
Substituting these values into the formula, we get:
P(A or B) = 5/36 + 1/6 - 1/18 = 10/36 = 5/18
Therefore, the probability of rolling a sum of 8 or doubles is 5/18.
c) P(BA) = P(AB)/P(A) = (2/36)/(5/36) = 2/5
Therefore, the probability of rolling doubles given that we rolled a sum of 8 is 2/5.