Question 1180016
<font color=black size=3>
Part (a)


W = event that you win the game
P(W) = probability of winning 
P(W) = 1/9
Since you win 1 week, but lose 8 weeks, so 1+8 = 9 weeks in total. 
P(L) = probability of losing
P(L) = 1-P(W)
P(L) = 1-1/9
P(L) = 9/9-1/9
P(L) = (9-1)/9
P(L) = 8/9


V(W) = net value of winning
V(W) = (amount won) - (cost of playing)
V(W) = 15*(number of players) - 15
V(W) = 15*(you + five others) - 15
V(W) = 15*(1+5) - 15
V(W) = 15*6 - 15
V(W) = 90 - 15
V(W) = 75
You net $75 if you win
V(L) = net value of losing
V(L) = -15
Meaning you lost $15.


We'll multiply the probability values with the corresponding net values
P(W)*V(W) = (1/9)*(75) = 8.33 approximately
P(L)*V(L) = (8/9)*(-15) = -13.33 approximately


Add up the results:
8.33+(-13.33) = -5


On average, you should expect to lose about $5 per week. Because this value is not zero, this means the game is not mathematically fair.


Answer: -5 dollars


=====================================================================
Part (b)


Let's say the probability of winning any given week is P(W) = x. 
That would mean P(L) = 1-x.


The net value of winning would be V(W) = 90-30 = 60, since you're spending $30 for the book.
Similarly, V(L) = 0-15-30 = -45. This is because you lose the $15 you bet in the game, on top of the $30 spent for the book. Overall, you lose $45.


We'll follow the same idea as before: multiply out the corresponding probability values with the net values, then add up the results


Multiply:
A = P(W)*V(W) = x*60 = 60x
B = P(L)*V(L) = (1-x)*(-45) = 45x-45


Add:
A+B = 60x+(45x-45) = 105x-45
This expression represents the expected value. 


We want the expected value expression to be positive. So we want it to be larger than 0. Let's solve for x
105x - 45 > 0
105x > 45
x > 45/105
x > (3*15)/(7*15)
x > 3/7
This indicates that the probability x must be larger than 3/7 in order to have a positive expected value. Keep in mind that the largest x can get is x = 1. So we could be more careful and write {{{3/7 < x <= 1}}} to express the full range of possible x values here.


Recall earlier that in part (a), the probability of winning was 1/9. This is without buying the book to improve your odds. We'll subtract 1/9 from 3/7 to get...
(3/7) - (1/9)
(27/63) - (7/63)
(27-7)/63
(27-7)/63
20/63
Going from a probability of 1/9 to 3/7 is an increase of 20/63. This is the amount the odds go up if you were to buy the book; assuming that we go for the smallest increase possible. To be more clear, the increase must be larger than this floor value to ensure that 105x-45 is positive.


In short, the book would have to increase your odds by at least 20/63 in order to get a positive expected value.


Answer: Increase by at least 20/63
Side note: 20/63 = 0.31746 approximately



Edit: My apologies, I did part (a) incorrectly earlier. I updated it to the correct solution however.  
</font>