Question

A group of 10 friends are playing a game of guessing the outcome of a tournament played among 8 teams (labeled team 1 to team 8) with a fixed schedule as follows.

In round 1, team 1 plays against teams 2, team 3 plays against team 4, team 5 plays against team 6, and team 7 plays against team 8. In round 2, the winner of (1,2) plays against the winner of (3,4), the winner of (5,6) plays against the winner of (7,8). In round 3, the winners of round 2 plays against each other.

The person who correctly predicts the outcome of all 3 rounds wins the game. The idea is that each person will put $2 forward (with a total of $20). What is the expected amount of money a person will lose or gain by playing this game?

Answer 1

ANSWER

Suppose, X be the amount of money a person gains.

In total, 4+2+1=7 matches will occur. In each match 10 persons put $2 each. So, in each match $20 is put. Hence, throughout the tournament $(7*20)=$140 will be put.

Predicting outcome of a game is independent of predicting outcome of other games.

So, probability that a person predicts all outcomes correctly is

$\small \left (\frac{1}{2} \right )^7=\frac{1}{128}$

A person has to pay $(7*2)=$14 throughout the tournament.

:. P(X = x) = { 128 if x = 140 – 14 127 if x = -14 128

if x = 126 + P(X = x) = { if x = -14

-.. E(X)= {* P(X = 1) = 178 * 126 - 12* * 14 = -12.90625

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