Alice, bob, and confucius are bored during recess, so they decide to play a new game. each of them puts a dollar in the pot, and each tosses a quarter. alice wins if the coins land all heads or all tails. bob wins if two heads and one tail land, and confucius wins if one head and two tails land. the quarters are fair, and the winner receives a net payment of $2 ($3 - $1 = $2), and the losers lose their $1. what is the probability that alice will win and the probability that she will lose? what is alice's expected payoff? what is the probability that confucius will win and the probability that he will lose? what is confucius's expected payoff? is this a zero-sum game? explain.

1/4 Alice wins and 3/34 she loses; Alice's expected payoff -$0.25; 3/8 Confucius wins and 5/8 he loses; Confucius's expected payoff $0.125; yes it is a zero sum game.

Step-by-step explanation:

First we create the sample space for flipping three coins:

HHH, HHT, HTH, THH, HTT, THT, TTH, TTT

There are a total of 8 outcomes.

Alice wins if all coins land either all heads or all tails; this is either HHH or TTT, giving her a 2/8 = 1/4 chance of winning.  This means she has a 6/8 = 3/4 chance of losing.

Alice's expected payoff is found by multiplying her chances of winning, 1/4, by the amount she wins, $2, and her chances of losing, 3/4, by the amount she loses, $1:

1/4(2) + 3/4(-1) = 2/4 + -3/4 = -1/4 = -$0.25.

Confucius wins if there is one head and two tails; this is HTT, THT, or TTH.  This gives him a 3/8 chance of winning, which means he has a 5/8 chance of losing.

Confucius's expected payoff is found by multiplying his chances of winning, 3/8, by the amount he wins, $2, and his chances of losing, 5/8, by the amount he loses, $1:

3/8(2) + 5/8(-1) = 6/8 + -5/8 = 1/8 = $0.125.

Bob wins if two heads and one tail land; this is either HHT, HTH, or THH.  This gives him a 3/8 chance of winning, which means he has a 5/8 chance of losing.

Bob's expected payoff is found by multiplying his chances of winning, 3/8, by the amount he wins, $2, and his chances of losing, 5/8, by the amount he loses, $1:

3/8(2) + 5/8(-1) = 6/8 + -5/8 = 1/8 = $0.125.

This makes the sum of the expected values

-$0.25+$0.125+$0.125 = 0; this makes it a zero sum game.