The problem in the image asks you to find the probability of each sum when rolling two dice. Here's a step-by-step explanation of how to approach this:

Step 1: Construct the sample space for rolling two dice.

When two dice are rolled, each die has 6 faces numbered 1 through 6. The total number of possible outcomes when rolling two dice is $6 \times 6 = 36$ because each die has 6 possible results.

Step 2: Determine the sums and their probabilities.

The possible sums when rolling two dice range from 2 to 12. Now, we need to determine how many ways each sum can be obtained.

• Sum = 2: (1,1) → 1 way
• Sum = 3: (1,2), (2,1) → 2 ways
• Sum = 4: (1,3), (2,2), (3,1) → 3 ways
• Sum = 5: (1,4), (2,3), (3,2), (4,1) → 4 ways
• Sum = 6: (1,5), (2,4), (3,3), (4,2), (5,1) → 5 ways
• Sum = 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) → 6 ways
• Sum = 8: (2,6), (3,5), (4,4), (5,3), (6,2) → 5 ways
• Sum = 9: (3,6), (4,5), (5,4), (6,3) → 4 ways
• Sum = 10: (4,6), (5,5), (6,4) → 3 ways
• Sum = 11: (5,6), (6,5) → 2 ways
• Sum = 12: (6,6) → 1 way

Step 3: Calculate the probability of each sum.

The probability of each sum is calculated as the number of ways to get that sum divided by the total number of outcomes (36). Let's list them:

• P(2) = $\frac{1}{36}$
• P(3) = $\frac{2}{36} = \frac{1}{18}$
• P(4) = $\frac{3}{36} = \frac{1}{12}$
• P(5) = $\frac{4}{36} = \frac{1}{9}$
• P(6) = $\frac{5}{36}$
• P(7) = $\frac{6}{36} = \frac{1}{6}$
• P(8) = $\frac{5}{36}$
• P(9) = $\frac{4}{36} = \frac{1}{9}$
• P(10) = $\frac{3}{36} = \frac{1}{12}$
• P(11) = $\frac{2}{36} = \frac{1}{18}$
• P(12) = $\frac{1}{36}$

These are the probabilities of rolling each sum when rolling two dice.

Do you need more details or have any questions?

Here are five related questions you might want to explore:

1. How would the probabilities change if you used two dice with different numbers of sides?
2. What is the probability of rolling a sum greater than or equal to 9?
3. What is the probability of rolling an odd sum with two dice?
4. How does the probability of getting a sum of 7 compare with the probabilities of other sums?
5. What is the probability of rolling a double (i.e., both dice showing the same number)?

Tip: The sum of 7 is the most likely outcome when rolling two dice because there are more combinations that result in a 7 than any other sum.