The game of dots is played by rolling a fair die and
receiving $1 for each dot showing on the top face of the
die. What cost should be set for each roll if the game is
to be considered a fair game?
You can get 1, 2, 3, 4, 5, or 6, with equal probability of getting any number, so the median/mean value would be the best choice. Can you calculate that?
To determine the fair cost for each roll of the game, we need to consider the expected value. The expected value is the average amount that a player can expect to win or lose per roll in the long run.
In this game, the player receives $1 for each dot showing on the top face of the die. Since the die is fair, it has six equally likely outcomes: showing 1 dot, 2 dots, 3 dots, 4 dots, 5 dots, or 6 dots.
To find the expected value, we'll calculate the average of the possible outcomes multiplied by their respective probabilities.
Expected value = (P(1) * $1) + (P(2) * $2) + (P(3) * $3) + (P(4) * $4) + (P(5) * $5) + (P(6) * $6)
Since the die is fair, each outcome has a probability of 1/6:
Expected value = (1/6 * $1) + (1/6 * $2) + (1/6 * $3) + (1/6 * $4) + (1/6 * $5) + (1/6 * $6)
Simplifying, we get:
Expected value = $1/6 + $2/6 + $3/6 + $4/6 + $5/6 + $6/6
Expected value = ($21/6)
Expected value = $3.50
Therefore, to make the game fair, the cost for each roll should be set at $3.50.