Answer:
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.