Question

One thousand tickets are sold at $1 each for a charity raffle. Tickets are to be drawn at random and cash prizes are to be awarded as follows: 1 prize of $100, 3 prizes of $50, and 5 prizes of $20. What is the expected value of this raffle if you buy 1 ticket?

Now suppose some benefactor agrees to give every player $1000, regardless of the outcome of the raffle. What is the expected value of the game? What general principle is at work?

I know that the expected value without the benefactor is -$0.65 and then with the benefactor the expected value is $999.35. I am, however, confused as to what general principle is at work. Could you help me understand what this general principle is?

Answer 1

Expected value E[x]=μ which represent the overall average value of the game
E[x]=∑ x*P(x) where
x=value of an outcome,
P(x)=probability of that outcome.

For given case, we sold 1000 tickets, out of which 1 prize of $100, 3 prizes of $50, and 5 prizes of $20.

So (1000-1-3-5 = 991) people will pay $1 (with probability of 991/1000 of losing) ,

and winners will get

(a) $100-$1=$99 with a probability of 1/1000 of winning).

(b) $50-$1=$49, with a probability of 3/1000 of winning).

(c) $20-$1=$19 with a probability of 5/1000 of winning).

Summing over the entire case
E[x]=(-1)*991/1000+(99)*1/1000+(49)*3/1000+(19)*5/1000

= -0.991 + 0.099 + 0.147 + 0.095

= -0.65 dollars