To determine if the game is fair, we need to calculate the expected value of the winnings. The game costs $1 to play, and the winnings are structured as follows:
- $5 for landing on red
- $3 for landing on blue
- $2 for landing on yellow
- $0 for any other outcome
Let’s assume the probabilities of landing on each color are as follows:
- Probability of red (P_red) = 1/3
- Probability of blue (P_blue) = 1/3
- Probability of yellow (P_yellow) = 1/3
Now, we can calculate the expected value (EV) of the game:
EV = (P_red * Winnings_red) + (P_blue * Winnings_blue) + (P_yellow * Winnings_yellow) + (P_other * Winnings_other)
Substituting in the values:
EV = (1/3 * 5) + (1/3 * 3) + (1/3 * 2) + (0 * P_other)
EV = (5/3) + (3/3) + (2/3) = 10/3
This results in an expected value of approximately $3.33.
Now, since the player has to pay $1 to play the game, we need to subtract this cost from the expected value:
Net EV = EV – Cost = (10/3) – 1 = (10/3) – (3/3) = 7/3
This results in a net expected value of approximately $2.33.
Since the expected value of approximately $2.33 is greater than $0, the game is not fair. A fair game should have an expected value of $0, meaning players would neither gain nor lose money in the long run. In this case, the player can expect to gain on average from each game played, which indicates the game favors the player.