Dice Probability Distribution

Let's break down the possible sums when you roll two dice:

1. Sum of 2: This can only happen if you roll a 1 on the first die and a 1 on the second die.

   • (1,1)

2. Sum of 3: This can happen in two ways:

   • (1,2) or (2,1)

3. Sum of 4:

   • (1,3), (3,1), (2,2)

4. Sum of 5:

   • (1,4), (4,1), (2,3), (3,2)

5. Sum of 6:

   • (1,5), (5,1), (2,4), (4,2), (3,3)

6. Sum of 7:

   • (1,6), (6,1), (2,5), (5,2), (3,4), (4,3)

7. Sum of 8:

   • (2,6), (6,2), (3,5), (5,3), (4,4)

8. Sum of 9:

   • (3,6), (6,3), (4,5), (5,4)

9. Sum of 10:

   • (4,6), (6,4), (5,5)

10. Sum of 11:

• (5,6), (6,5)

11. Sum of 12:

• (6,6)

Since there are $6 \times 6 = 36$ possible outcomes when you roll two dice, we can find the probabilities by dividing the number of ways to get each sum by 36:

1. Sum of 2: $\frac{1}{36}$
2. Sum of 3: $\frac{2}{36}$ = $\frac{1}{18}$
3. Sum of 4: $\frac{3}{36}$ = $\frac{1}{12}$
4. Sum of 5: $\frac{4}{36}$ = $\frac{1}{9}$
5. Sum of 6: $\frac{5}{36}$
6. Sum of 7: $\frac{6}{36}$ = $\frac{1}{6}$
7. Sum of 8: $\frac{5}{36}$
8. Sum of 9: $\frac{4}{36}$ = $\frac{1}{9}$
9. Sum of 10: $\frac{3}{36}$ = $\frac{1}{12}$
10. Sum of 11: $\frac{2}{36}$ = $\frac{1}{18}$
11. Sum of 12: $\frac{1}{36}$

This distribution is triangular in shape with the peak at 7, which has the highest probability of occurring.

I'm sorry, I cannot create visual plots directly within this text-based interface. However, I can guide you on how to create one:

1. Horizontal Axis (X-axis): This represents the possible sums ranging from 2 to 12.

2. Vertical Axis (Y-axis): This represents the probability of each sum.

3. Bars: For each possible sum, you'd plot a vertical bar whose height corresponds to its probability.

Here's a textual representation of the distribution:

mathematicaCopy code
Sum  Probability (in 36ths)
2    |
3    ||
4    |||
5    ||||
6    |||||
7    ||||||
8    |||||
9    ||||
10   |||
11   ||
12   |

In an actual plotted graph:

• The height of the bars would correspond to the probabilities.
• The bars for sums 2 and 12 would be the shortest since each has only 1/36 chance.
• The bar for sum 7 would be the tallest since it has a 6/36 (or 1/6) chance.

You can use software like Microsoft Excel, Google Sheets, or any data visualization tool to easily create this histogram/bar chart.