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.

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

1. 3.4/4

2. 0.8/7

3. 0.5 x 3

4. 0.9 x 6

5. 10.8

6. 2.35

7. 0.567

8. 9.35

9. 0.02

10. 0.06

11. 0.16

12. 0.42

13. 0.63

14. 5.6

15. 8.1

16. 9.9

17. 0.28

18. 0.48

19. 0.27

20. 0.49

21. 0.45

22. 0.72

23. 0.9

24. 0.88

25. 0.75

26. 0.16

27. 0.328

28. 0.495

29. 0.72

30. 1.068