# Dice Probability Distribution

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

**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)

**Sum of 3**: This can happen in two ways:- (1,2) or (2,1)

**Sum of 4**:- (1,3), (3,1), (2,2)

**Sum of 5**:- (1,4), (4,1), (2,3), (3,2)

**Sum of 6**:- (1,5), (5,1), (2,4), (4,2), (3,3)

**Sum of 7**:- (1,6), (6,1), (2,5), (5,2), (3,4), (4,3)

**Sum of 8**:- (2,6), (6,2), (3,5), (5,3), (4,4)

**Sum of 9**:- (3,6), (6,3), (4,5), (5,4)

**Sum of 10**:- (4,6), (6,4), (5,5)

**Sum of 11**:

- (5,6), (6,5)

**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:

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

**Horizontal Axis (X-axis)**: This represents the possible sums ranging from 2 to 12.**Vertical Axis (Y-axis)**: This represents the probability of each sum.**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:

`mathematica````
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.