Question

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. (2 points)

Answer 1

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:

\[
\boxed{\frac{1}{72}}
\]