Final answer:
To set the expected value of the game to zero when flipping a fair coin that pays $1 for every heads and ends on tails, the initial fee should be set at $1. The sum of the expected gains from an infinite series of flips is $1, thus balancing the cost to play.
Step-by-step explanation:
The question asks for the expected value of a coin-flipping game to be zero when considering a fair coin. To calculate the expected value of the game, we need to determine the average amount that would be won over many repetitions of the game.
In this game, you flip a coin and receive $1 for every head flipped, with the game ending when a tail is flipped. A fair coin has a 50% (0.5 probability) chance of landing on heads or tails. To find the expected value (EV), we can set up an equation summing the probability-weighted outcomes.
- Probability of getting heads once: 0.5 (earn $1)
- Probability of getting heads twice: 0.25 (earn $2)
- ... and so on.
The sum is an infinite series, which is the sum of 1/2 + 1/4 + 1/8 + ..., where each term represents the probability of continuing the game and earning another dollar. This is equal to 1 when summed to infinity, meaning the expected gain from flipping the coin until tails appear is $1. Therefore, to set the expected value of the game to zero, the initial fee should be set at $1.
EV for one round of the game is $0 if the entrance fee is $1, since:
EV = (Amount Won) - (Cost to Play) = $1 - $1 = $0.