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Sagot :
Answer:
1) The expected value of the "winnings" is $(-0.97[tex]\overline 2[/tex])
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[tex]\overline 2[/tex]
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[tex]\overline 6[/tex]
For 3, we have 1/36 × (20 - 5) = 0.41[tex]\overline 6[/tex]
For 4, we have 1/18 × (20 - 5) = 0.8[tex]\overline 3[/tex]
For 5, we have 1/18 × (10 - 5) = 0.2[tex]\overline 7[/tex]
For 6, we have 1/12 × (10 - 5) = 0.41[tex]\overline 6[/tex]
For 7, we have 1/12 × (10 - 5) = 0.41[tex]\overline 6[/tex]
For 8, we have 1/12 × (10 - 5) = 0.41[tex]\overline 6[/tex]
For 9, we have 1/18 × (-20 - 5) = -1.3[tex]\overline 8[/tex]
For 10, we have 1/18 × (-20 - 5) = -1.3[tex]\overline 8[/tex]
For 11, we have 1/36 × (-20 - 5) = -0.69[tex]\overline 4[/tex]
For 12, we have 1/36 × (-25 - 5) = -0.69[tex]\overline 4[/tex]
The expected value of the winnings is given as follows;
0.41[tex]\overline 6[/tex] + 0.41[tex]\overline 6[/tex] + 0.8[tex]\overline 3[/tex] + 0.8[tex]\overline 3[/tex] + 0.8[tex]\overline 3[/tex] + 0.41[tex]\overline 6[/tex] + 0.41[tex]\overline 6[/tex] + -1.3[tex]\overline 8[/tex] -1.3 - 0.69[tex]\overline 4[/tex] - 0.69[tex]\overline 4[/tex] = -0.97[tex]\overline 2[/tex]
Therefore, the expected value = $-0.97[tex]\overline 2[/tex], which is one is expected to lose $0.97[tex]\overline 2[/tex]
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[tex]\overline 2[/tex]
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