To find the probability that Nikki rolls a sum of 4 with two dice and then rolls a sum of 7 with the same or another pair of dice, we will calculate these probabilities separately and then multiply them together.
Step 1: Probability of rolling a sum of 4
When rolling two six-sided dice, the possible pairs (d1, d2) that result in a sum of 4 are:
- (1, 3)
- (2, 2)
- (3, 1)
Counting the pairs, we find there are 3 combinations that sum to 4.
The total number of outcomes when rolling two dice is:
\[ 6 \times 6 = 36 \]
Thus, the probability \( P(4) \) of rolling a sum of 4 is:
\[ P(4) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{3}{36} = \frac{1}{12} \]
Step 2: Probability of rolling a sum of 7
Next, we need to determine the number of combinations that yield a sum of 7. The possible pairs (d1, d2) that result in a sum of 7 are:
- (1, 6)
- (2, 5)
- (3, 4)
- (4, 3)
- (5, 2)
- (6, 1)
Counting the pairs, we find there are 6 combinations that sum to 7.
Thus, the probability \( P(7) \) of rolling a sum of 7 is:
\[ P(7) = \frac{6}{36} = \frac{1}{6} \]
Step 3: Combined probability of rolling a sum of 4 then a sum of 7
Since rolling the two pairs of dice are independent events, we can multiply the probabilities:
\[ P(\text{sum of 4 and then sum of 7}) = P(4) \times P(7) = \frac{1}{12} \times \frac{1}{6} = \frac{1}{72} \]
Thus, the final probability that Nikki rolls a sum of 4 and then rolls a sum of 7 is:
\[ \boxed{\frac{1}{72}} \]