To find the probability that Nikki rolls a sum of 4 with two dice and then rolls a sum of 7, we can break it down into two parts: calculating the probability of each event and then multiplying those probabilities together.
Step 1: Probability of rolling a sum of 4
To roll a sum of 4 with two dice, we can list the different combinations that result in this sum:
- (1, 3)
- (2, 2)
- (3, 1)
So there are 3 combinations that give a sum of 4.
The total number of possible outcomes when rolling two dice is \(6 \times 6 = 36\).
Thus, the probability of rolling a sum of 4 is: \[ P(\text{sum of 4}) = \frac{\text{Number of combinations for sum of 4}}{\text{Total outcomes}} = \frac{3}{36} = \frac{1}{12} \]
Step 2: Probability of rolling a sum of 7
Next, we calculate the probability of rolling a sum of 7. The combinations for a sum of 7 are:
- (1, 6)
- (2, 5)
- (3, 4)
- (4, 3)
- (5, 2)
- (6, 1)
There are 6 combinations that yield a sum of 7.
Thus, the probability of rolling a sum of 7 is: \[ P(\text{sum of 7}) = \frac{\text{Number of combinations for sum of 7}}{\text{Total outcomes}} = \frac{6}{36} = \frac{1}{6} \]
Step 3: Combined probability
Now, to find the probability that Nikki first rolls a sum of 4 and then a sum of 7, we multiply the probabilities of the two independent events:
\[ P(\text{sum of 4 and then sum of 7}) = P(\text{sum of 4}) \times P(\text{sum of 7}) \] \[ = \frac{1}{12} \times \frac{1}{6} = \frac{1}{72} \]
Final Result
The probability that Nikki rolls a sum of 4 with the two dice and then rolls a sum of 7 with the two dice is:
\[ \boxed{\frac{1}{72}} \]