Another Math questions is about dices?

Question

When a pair of dice is rolled, the total will range from 2 (1,1) to 12 (6,6). It is a fact that some numbers will occur more frequently than others as the dice are rolled over and over.

Why will some numbers come up more frequently than others?
Each die has six sides numbered from 1 to 6. How many possible ways can a number be rolled? In other words, we can roll (2,3) or (3,2) or (6,1) and so on. What are the total (x,y) outcomes that can occur?
How might you then estimate the percentage of the time a particular number will come up if the dice are rolled over and over?
Once these percentages have been calculated, how might the mean value of the all the numbers thrown be determined?

Answer 1

The first answer is misleading if you want to calculate odds. For example, while there is only one way to roll a 2 (1 on each die) there are two ways to roll a 3 (1 on the first die and 2 on the second die OR 2 on the first die and 1 on the second die). As a result 3 will come up twice as often as 2.

Since each die has six sides, there are 36 (6x6) possible combinations.

1 way to roll a 2
2 ways to roll a 3
3 ways to roll a 4
4 ways to roll a 5
5 ways to roll a 6
6 ways to roll a 7
5 ways to roll an 8
4 ways to roll a 9
3 ways to roll a 10
2 ways to roll an 11
1 way to roll a 12

To get the percentage for any given number, divide the number of ways to roll it by 36. For example, a seven will be rolled an average of 6 out of every 36 rolls, or 16.667% of the time.

The mean value for all rolls is 7.

The formula to calculate is to multiply all each possible result by the number of ways to roll it, sum all the products for those multiplications, and divide the total by 36.

Answer 2

Well the most probable outcomes are 6, 7, or 8 since they each are possible in 3 different combinations.
There are 21 possible combinations to be rolled.
For each numerical outcome there are the following number of combinations
2 - 1 = 1 & 1
3 - 1 = 2 & 1
4 - 2 = 1 & 3 - 2 & 2
5 - 2 = 1 & 4 - 2 & 3
6 - 3 = 1 & 5 - 2 & 4 - 3 & 3
7 - 3 = 1 & 6 - 2 & 5 - 3 & 4
8 - 3 = 2 & 6 - 3 & 5 - 4 & 4
9 - 2 = 3 & 6 - 4 & 5
10 - 2 = 4 & 6 - 5 & 5
11 - 1 = 5 & 6
12 - 1 = 6 & 6

3 possible combinations equals roughly 14.3%
2 combinations = 9.5%
1 combinations = 4.7%
(14.3 x 3) + (9.5 x 4) + (4.7 x 4) = 99.7 (like I said, rough percentages)

Answer 3

well, you just gotta go through the combinations.

2....1,1 -- 1/36 2.7%
3....1,2/2,1 -- 1/18 5.5%
4....1,3/2,2/3,1 -- 1/12 8.3%
5....2,3/4,1/4,1/3,2 -- 1/9 11.1%
6....1,5/2,4/3,3/4,2/5,1 -- 5/36 13.8%
7....1,6/2,5/3,4/4,3/5,2/6,1 -- 1/6 16.6%
8-12 just go in reverse.

Just take how many combinations you want and divide by 36 to get your percentage.