Answer:
a. The expected value ( or mean ) is -1.
b. It is not a fair game.
Step-by-step explanation:
a.
First we look into all possible values for a random variable X, that is defined as:
X = The sum of the number on both dice.
Next there is a table, that shows us the frequency of a certain sum in all possible combinations:
1 2 3 4 5 6
1 2 3 4 5 6 7
2 3 4 5 6 7 8
3 4 5 6 7 8 9
4 5 6 7 8 9 10
5 6 7 8 9 10 11
6 7 8 9 10 11 12
You can see that the sum 2 appears only one time, as opposed to the sum 7, that appears 6 times.
Now, let's create a random variable called Y, so that
Y = The sum of the number on both dice minus 8.
So if we count the frequency of all possible sums minus 8, we get the following:
![\left[\begin{array}{ccc}Y&P(Y=y)\\-6&1/36 \\-5&2/36\\-4&3/36\\-3&4/36\\-2&5/36\\-1&6/36\\0&5/36\\1&4/36\\2&3/36\\3&2/36\\4&1/36\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7DY%26P%28Y%3Dy%29%5C%5C-6%261%2F36%20%5C%5C-5%262%2F36%5C%5C-4%263%2F36%5C%5C-3%264%2F36%5C%5C-2%265%2F36%5C%5C-1%266%2F36%5C%5C0%265%2F36%5C%5C1%264%2F36%5C%5C2%263%2F36%5C%5C3%262%2F36%5C%5C4%261%2F36%5Cend%7Barray%7D%5Cright%5D)
So from now on, we just need to calculate the expected value:
![E(Y) = (-6)\frac{1}{36} + (-5)\frac{2}{36} \cdots + 4\frac{1}{36} = \sum_{i=-6} ^{4}iP(Y=i) = -1](https://tex.z-dn.net/?f=E%28Y%29%20%3D%20%28-6%29%5Cfrac%7B1%7D%7B36%7D%20%2B%20%28-5%29%5Cfrac%7B2%7D%7B36%7D%20%5Ccdots%20%2B%204%5Cfrac%7B1%7D%7B36%7D%20%3D%20%5Csum_%7Bi%3D-6%7D%20%5E%7B4%7DiP%28Y%3Di%29%20%3D%20-1)
b.
From this, as the expected value isn't equal to zero, that means it's a unfair game or a biased game, it's so that one side always triumph over the other.