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You enter an online cooking tournament. The entry fee is $16, and the entrants make the recipes at home and vote for their favorite, which cannot be their own recipe. The top 3 finishers win money. First place wins $58, and second place wins $47. Third place wins $5. You give yourself an X chance of finishing in the top 3, and if you finish top 3, then you have an equal chance of finishing first, second, or third. What is X such that entering this tournament is a fair gamble for you. A fair gamble is one where the expected value of entering the tournament is $0.

Respuesta :

batolisis

Answer:

X =  0.5714

Step-by-step explanation:

entry fee = $16

price for first place = $58.   Net win = $58 - $16 = $42

price for second place = $47.  Net win = $47 - $16 = $21

price for third place = $5.  Net win = $5 - $16 = -$11 ( loss ) hence third position is not a fair gamble

Net loss for the rest =  -$16

Given that

P(1st place ) = P( 2nd place ) = P(third place ) =  X/3

P( rest ) = 1 - X

Hence the E(profit )

= 42 * X/3 + 37 * X/3 - 11 * X/3 - 16* ( 1 - X )

= 42X / 3  + 37X / 3 -  11X / 3 - 16 + 16X

= 68 X / 3 + 16X - 16

= 84X / 3 - 16 =  28x - 16

hence : 28X - 16 = $0    is a fair gamble

X = 16 / 28 = 0.5714

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