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?
There is a 1/6 chance of getting any of the numbers 1, 2, 3, 4, 5 or 6.
The median of these scores is 3.5, meaning that by chance 50% of the time you would get above 3.5 and 50% of the time you would get below.
Using that reasoning. it would be fair to charge $3.50 a roll. However, if I were running the game, I would charge at least $4, so I could make a profit.
To determine the fair cost for each roll in the game of dots, we can start by analyzing the possible outcomes of rolling a fair die.
A fair die has six equally likely outcomes - numbers 1 through 6. In this game, we receive $1 for each dot showing, which corresponds to the number rolled on the die.
Let's consider the expected value, or average winnings, for a single roll of the die in this game. We can calculate it by adding up the possible winnings and dividing by the number of outcomes:
Expected Value = ((1 * P(1)) + (2 * P(2)) + (3 * P(3)) + (4 * P(4)) + (5 * P(5)) + (6 * P(6))) / 6
To calculate this rate, we need to determine the probability of rolling each number. Since we have a fair die, the probability of rolling each number is 1/6.
Therefore, the expected value simplifies to:
Expected Value = (1/6 + 2/6 + 3/6 + 4/6 + 5/6 + 6/6) / 6
Expected Value = (21/6) / 6
Expected Value = 21/36
Expected Value = 7/12
Now, we can consider this expected value as the average winnings for each roll. To make the game fair, the cost per roll should be set to this expected value.
Therefore, the fair cost for each roll in the game of dots should be $7/12, or approximately $0.58 (rounded to the nearest cent).