Answer:
1) The expected value of the "winnings" is $(-0.97)
2) The variance for the "winnings" is $0.57966
3) The standard deviation for the "winnings" is$0.761354
4) The game is not a fair game because one is expected to lose $0.97
Step-by-step explanation:
1) The probability of having a sum of 2 = 1/6×1/6 = 1/36
The probability of having a sum of 3 = 1/6×1/6 = 1/36
The probability of having a sum of 4 = 1/6×1/6 + 1/6×1/6 = 1/18
The probability of having a sum of 5 = 1/6×1/6 + 1/6×1/6 = 1/18
The probability of having a sum of 6 = 1/6×1/6 + 1/6×1/6 + 1/6×1/6 = 1/12
The probability of having a sum of 7 = 1/6×1/6 + 1/6×1/6 + 1/6×1/6 = 1/12
The probability of having a sum of 8 = 1/6×1/6 + 1/6×1/6 + 1/6×1/6 = 1/12
The probability of having a sum of 9 = 1/6×1/6 + 1/6×1/6 = 1/18
The probability of having a sum of 10 = 1/6×1/6 + 1/6×1/6 = 1/18
The probability of having a sum of 11 = 1/6×1/6 = 1/36
The probability of having a sum of 12 = 1/6×1/6 = 1/36
The values are;
For 2, we have 1/36 × (20 - 5) = 0.41
For 3, we have 1/36 × (20 - 5) = 0.41
For 4, we have 1/18 × (20 - 5) = 0.8
For 5, we have 1/18 × (10 - 5) = 0.2
For 6, we have 1/12 × (10 - 5) = 0.41
For 7, we have 1/12 × (10 - 5) = 0.41
For 8, we have 1/12 × (10 - 5) = 0.41
For 9, we have 1/18 × (-20 - 5) = -1.3
For 10, we have 1/18 × (-20 - 5) = -1.3
For 11, we have 1/36 × (-20 - 5) = -0.69
For 12, we have 1/36 × (-25 - 5) = -0.69
The expected value of the winnings is given as follows;
0.41 + 0.41
+ 0.8
+ 0.8
+ 0.8
+ 0.41
+ 0.41
+ -1.3
-1.3 - 0.69
- 0.69
= -0.97
Therefore, the expected value = $-0.97, which is one is expected to lose $0.97
2) Using Microsoft Excel, we have;
The variance for the "winnings", σ² = $0.57966
3) The standard deviation for the "winnings" = √σ² = √(0.57966) ≈ $0.761354
4) The game is not a fair game because one is expected to lose $0.97