Two dice are rolled. Determine the probability of the following. ("Doubles" means both dice show the same number. Enter your probabilities as fractions.)
Rolling an even sum OR doubles
There are a total of 36 possible outcomes when rolling two dice (6 possible outcomes for each of the first die multiplied by 6 possible outcomes for the second die).
1. Rolling an even sum:
There are 18 outcomes that result in an even sum:
(2,2), (2,4), (2,6), (4,2), (4,4), (4,6), (6,2), (6,4), (6,6)
So the probability of rolling an even sum is 18/36 = 1/2.
2. Rolling doubles:
There are 6 outcomes that result in doubles:
(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)
So the probability of rolling doubles is 6/36 = 1/6.
To find the probability of rolling an even sum OR doubles, we need to consider the outcomes that fall into either category but not both. Since rolling doubles also results in an even sum, we need to subtract the probability of rolling doubles from the probability of rolling an even sum to avoid double counting:
Probability = Probability of rolling an even sum + Probability of rolling doubles - Probability of rolling both an even sum and doubles
Probability = 1/2 + 1/6 - 1/36
Probability = 18/36 + 6/36 - 1/36
Probability = 23/36
Therefore, the probability of rolling an even sum OR doubles is 23/36.