Answer:
(a)$67
(b)You are expected to win 56 Times
(c)You are expected to lose 44 Times
Step-by-step explanation:
The sample space for the event of rolling two dice is presented below
![(1,1), (2,1), (3,1), (4,1), (5,1), (6,1)\\(1,2), (2,2), (3,2), (4,2), (5,2), (6,2)\\(1,3), (2,3), (3,3), (4,3), (5,3), (6,3)\\(1,4), (2,4), (3,4), (4,4), (5,4), (6,4)\\(1,5), (2,5), (3,5), (4,5), (5,5), (6,5)\\(1,6), (2,6), (3,6), (4,6), (5,6), (6,6)](https://tex.z-dn.net/?f=%281%2C1%29%2C%20%282%2C1%29%2C%20%283%2C1%29%2C%20%284%2C1%29%2C%20%285%2C1%29%2C%20%286%2C1%29%5C%5C%281%2C2%29%2C%20%282%2C2%29%2C%20%283%2C2%29%2C%20%284%2C2%29%2C%20%285%2C2%29%2C%20%286%2C2%29%5C%5C%281%2C3%29%2C%20%282%2C3%29%2C%20%283%2C3%29%2C%20%284%2C3%29%2C%20%285%2C3%29%2C%20%286%2C3%29%5C%5C%281%2C4%29%2C%20%282%2C4%29%2C%20%283%2C4%29%2C%20%284%2C4%29%2C%20%285%2C4%29%2C%20%286%2C4%29%5C%5C%281%2C5%29%2C%20%282%2C5%29%2C%20%283%2C5%29%2C%20%284%2C5%29%2C%20%285%2C5%29%2C%20%286%2C5%29%5C%5C%281%2C6%29%2C%20%282%2C6%29%2C%20%283%2C6%29%2C%20%284%2C6%29%2C%20%285%2C6%29%2C%20%286%2C6%29)
Total number of outcomes =36
The event of rolling a 5 or a 6 are:
![(5,1), (6,1)\\ (5,2), (6,2)\\( (5,3), (6,3)\\ (5,4), (6,4)\\(1,5), (2,5), (3,5), (4,5), (5,5), (6,5)\\(1,6), (2,6), (3,6), (4,6), (5,6), (6,6)](https://tex.z-dn.net/?f=%285%2C1%29%2C%20%286%2C1%29%5C%5C%20%285%2C2%29%2C%20%286%2C2%29%5C%5C%28%20%285%2C3%29%2C%20%286%2C3%29%5C%5C%20%285%2C4%29%2C%20%286%2C4%29%5C%5C%281%2C5%29%2C%20%282%2C5%29%2C%20%283%2C5%29%2C%20%284%2C5%29%2C%20%285%2C5%29%2C%20%286%2C5%29%5C%5C%281%2C6%29%2C%20%282%2C6%29%2C%20%283%2C6%29%2C%20%284%2C6%29%2C%20%285%2C6%29%2C%20%286%2C6%29)
Number of outcomes =20
Therefore:
P(rolling a 5 or a 6) ![=\dfrac{20}{36}](https://tex.z-dn.net/?f=%3D%5Cdfrac%7B20%7D%7B36%7D)
The probability distribution of this event is given as follows.
![\left|\begin{array}{c|c|c}$Amount Won(x)&-\$1&\$2\\&\\P(x)&\dfrac{16}{36}&\dfrac{20}{36}\end{array}\right|](https://tex.z-dn.net/?f=%5Cleft%7C%5Cbegin%7Barray%7D%7Bc%7Cc%7Cc%7D%24Amount%20Won%28x%29%26-%5C%241%26%5C%242%5C%5C%26%5C%5CP%28x%29%26%5Cdfrac%7B16%7D%7B36%7D%26%5Cdfrac%7B20%7D%7B36%7D%5Cend%7Barray%7D%5Cright%7C)
First, we determine the expected Value of this event.
Expected Value
![=(-\$1\times \frac{16}{36})+ (\$2\times \frac{20}{36})\\=\$0.67](https://tex.z-dn.net/?f=%3D%28-%5C%241%5Ctimes%20%5Cfrac%7B16%7D%7B36%7D%29%2B%20%28%5C%242%5Ctimes%20%5Cfrac%7B20%7D%7B36%7D%29%5C%5C%3D%5C%240.67)
Therefore, if the game is played 100 times,
Expected Profit =$0.67 X 100 =$67
If you play the game 100 times, you can expect to win $67.
(b)
Probability of Winning ![=\dfrac{20}{36}](https://tex.z-dn.net/?f=%3D%5Cdfrac%7B20%7D%7B36%7D)
If the game is played 100 times
Number of times expected to win
![=\dfrac{20}{36} \times 100\\=56$ times](https://tex.z-dn.net/?f=%3D%5Cdfrac%7B20%7D%7B36%7D%20%5Ctimes%20100%5C%5C%3D56%24%20times)
Therefore, number of times expected to loose
= 100-56
=44 times