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) write it the easy way

Answer 1

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. (1, 3)
2. (2, 2)
3. (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. (1, 6)
2. (2, 5)
3. (3, 4)
4. (4, 3)
5. (5, 2)
6. (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:

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