Nikki is rolling two dice. What is the probability that she rolls a sum of 4 with the two dice, then rolls a sum of 7 with the two dice? Show all your work.
To find the probability of rolling a sum of 4 with two dice, we first need to determine the number of ways the sum of 4 can be obtained.
There are three ways to roll a sum of 4:
- Rolling a 1 on the first die and a 3 on the second die
- Rolling a 2 on the first die and a 2 on the second die
- Rolling a 3 on the first die and a 1 on the second die
Since there are a total of 36 possible outcomes when rolling two dice (6 possibilities for the first die multiplied by 6 possibilities for the second die), the probability of rolling a sum of 4 is 3/36 or 1/12.
Next, we need to find the probability of rolling a sum of 7 with the two dice.
There are six ways to roll a sum of 7:
- (1,6)
- (2,5)
- (3,4)
- (4,3)
- (5,2)
- (6,1)
So, the probability of rolling a sum of 7 is 6/36 or 1/6.
To find the probability of rolling a sum of 4 and then a sum of 7, we multiply the individual probabilities together:
(1/12) * (1/6) = 1/72
Therefore, the probability of rolling a sum of 4 with the two dice, then rolling a sum of 7 with the two dice is 1/72.