Araffle has 10000 tickets. the grand prize wins a motorcycle worth $9000. additionally, there is one ticket that pays $2000 and another one that pays $1000. the remaining tickets will win nothing. if you have a ticket, what is the expected payoff for that ticket? a)$.90 b)$1.00 c)$1.20 d)$1.80

Choice C) $1.20



Work Shown:

Define the following events:
A = event of winning the motorcycle (valued at $9000)
B = event of winning $2000
C = event of winning $1000
D = event of winning nothing ($0)

The probability for each event is...
P(A) = 1/10000 = 0.0001
P(B) = 1/10000 = 0.0001
P(C) = 1/10000 = 0.0001
P(D) = 1 - (P(A)+P(B)+P(C))
P(D) = 1 - (0.0001+0.0001+0.0001)
P(D) = 0.9997
Or another way to see this is that there are 10000-3 = 9997 tickets that don't win anything so 9997/10000 = 0.9997

The net value for each event is
V(A) = 9000
V(B) = 2000
V(C) = 1000
V(D) = 0

Multiply the probabilities and the net values
P(A)*V(A) = 0.0001*9000 = 0.9
P(B)*V(B) = 0.0001*2000 = 0.2
P(C)*V(C) = 0.0001*1000 = 0.1
P(D)*V(D) = 0.9997*0 = 0

Then add up the results: 0.9+0.2+0.1+0 = 1.20

The expected value is 1.20

So that's why the answer is choice C

Note: this is assuming that the ticket cost $0 (or that a friend bought the ticket to be given as a gift to you)

Another note: if the ticket costs $1.20, then this is a fair game. Anything more, and the game would be unfair in favor of the house.