Final answer:
The expected value for Bill in the coin-tossing game is -$0.25, meaning he loses 25 cents on average per game. For Larry, the expected value is $0.25, which means he gains 25 cents on average per game.
Step-by-step explanation:
To calculate the expected value for Bill in the game where he tosses 2 coins with Larry, we need to consider both potential outcomes and their associated probabilities. Since the coins are fair, the probability of getting double heads (HH) is (1/2) * (1/2) = 1/4, and the probability of any other outcome (HT, TH, TT) is 3/4.
The expected value for Bill is calculated by multiplying the outcome by its respective probability and then adding these together. For a HH outcome, Bill would lose $4, while for any other outcome, he would gain $1. So for Bill, EV = (1/4) * (-$4) + (3/4) * $1 = -$1 + $0.75 = -$0.25.
For Larry, the situation is reversed, as he gains $4 if HH comes up and loses $1 otherwise. Larry's EV = (1/4) * $4 + (3/4) * (-$1) = $1 - $0.75 = $0.25.
The expected value indicates the average amount one would win or lose per game if the game is played many times. Bill is expected to lose 25 cents on average for each game played, while Larry is expected to gain 25 cents.