Question

You and your friend are rolling number cubes. Each cube has 6 sides with the numbers 1 to 6. If a sum of 7 is rolled, your friend gets a point. However, you get a point if either one of the sums listed below is rolled. Which pair of numbers would make the game fair?

Answer 1

Answer: Hello mate!

Ok, the 6 sides cubes can be thinked as two normal dice.

A fair game is that game where both players have the same chance to win, first, let's calculate the situations where you friend win:

this are, if the cubes have numers between 1 and 6:

1 + 6

6 + 1

2 + 5

5 + 2

3 + 4

4 + 3

where 1 + 6 is different to 6 + 1 because in the first one you rolled a 1 and your friend a 6, and in the second one you rolled a 6 and your friend a 1, so are different events.

Then you can see that you friend has 6 combinations where he wins.

So we need to find two numbers that between them, have the same number of combinations.

I can think, for example in 4 and 10:

the combinations for 4 are:

2 + 2

3 + 1

1 + 3

a total of 3 combinations:

for 10 are:

5 + 5

4 + 6

6+ 4

also 3 combinations.

So if you win a point if the sum is 4 or 10, you also have 6 combinations where you win, then the game is fair

Now, suppose the first number is 5, let's find the second:

the combinatios for 5 are:

4 + 1

1+ 4

2 + 3

3 + 2

a total of 4 combinations, this means that we need to find another number that only has two combinations, wich can be, for example, 3

the combinations for 3 are:

1 + 2

2 + 1

2 combinations, then if your numbers are 3 and 5, you have 6 combinations where you win, and the game is fair.

Answer 2

You don't list the "sums listed below" but I can still help you.
There are 6 ways to get a sum of 7 using a pair of dice:
1+6
6+1    You might be asking why 2+5 isn't the same as 5+2
2+5      Imagine the dice are different colors.  Is a red 2 and green 5 the
5+2          same as a green 2 and red 5?  (No, it's not)
3+4
4+3
You need two numbers that also provide 6 ways to win.  One pair that would work is 11 and 9   There are two ways to make 11 (5+6) and (6+5)  and there are four ways to make 9 (3+6), (6+3) (4+5) and (5+4) - that's a total of six ways.  So to balance your friend's 7 - you could go for sums of 11's and 9's.  Check the choices you were given and find the numbers (it could be three numbers like 2,3, and 12) that give you 6 ways to win.
Here are the numbers and their ways
Sum of the dice                          Ways to get it
2                                                      1
3                                                      2
4                                                      3
5                                                      4
6                                                      5
7                                                      6
8                                                      5
9                                                      4
10                                                    3
11                                                    2
12                                                    1