Question

The entry fee is $11, and the entrants make the recipes at home and vote for their favorite, which cannot be their own recipe. The top 2 finishers win money. First place wins $58, and second place wins $48. You give yourself an X chance of finishing in the top 2, and if you finish top 2, then you have an equal chance of finishing first or second.

Required:
What is x such that entering this tournament is a fair gamble for you as an expected value maximizer?

Answer 1

The value is
`X =  0.1719`

Explanation:

From the question we are told that

The entry fee is
`a =  \$ 11`

The outcome for first place is
`b =  \$ 58`

The outcome for second place is
`c =  \$ 48`

The outcome for losing is
`d =  - \$ 11`

The chance of finishing in the top two is X

Given that there are equal chance of finishing first or second, then the chance of finishing first is
`(X)/(2)`

and the chance of finishing second is
`(X)/(2)`

Then the chance of losing(i.e not finishing in the first two) is
`1 - X`

given that the game is a fair gamble for the player as an expected maximizer then it mean is that the expected value of entering the tournament E(X) = 0

So

`E(X) =  b *  (X)/(2)  +  c  *  (X)/(2) *  d  *  ( 1 - X)`

=>
`0 =  58 *  (X)/(2)  +  48  *  (X)/(2) *  -11  *  ( 1 - X)`

=>
`0 =  (58X)/(2)  +   (48X)/(2) *  -11  + 11X)`

=>
`X =  (64)/(11)`

=>
`X =  0.1719`