Final answer:
The expected value (EV) of the game when tossing two coins is calculated based on the probabilities of the outcomes. Each outcome (2 heads, 1 head, no heads) has a probability of 1/4, 1/2, and 1/4 respectively. The calculated EV is 0.25 points per turn, indicating a slightly favorable game for the player.
Step-by-step explanation:
To determine if a game is fair, one must calculate the expected value (EV) of the game's outcomes. In this game, you toss two fair coins and score points based on the number of heads that show up. There are three possible outcomes:
- 2 heads – Win 2 points (H-H)
- 1 head – Win 1 point (H-T or T-H)
- 0 heads – Lose 3 points (T-T)
The probability of each outcome is as follows:
- 2 heads (H-H): The probability is 1/4 since there is only one way to get two heads out of four possible outcomes (HH, HT, TH, TT).
- 1 head (H-T or T-H): The probability is 1/2 since there are two ways to get one head out of four possible outcomes.
- 0 heads (T-T): The probability is 1/4, similar to the two heads scenario.
Now, we calculate the expected value using the probabilities and the points associated with each outcome:
- (2 points * 1/4 chance) + (1 point * 1/2 chance) - (3 points * 1/4 chance)
- (2/4) + (1/2) - (3/4)
- (0.5) + (0.5) - (0.75)
- (1) - (0.75)
- 0.25
Therefore, the expected value is 0.25 points per turn, which suggests that the game is slightly in the player's favor, making it a potentially profitable game in the long run.