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?
Expected value = (1+2+3+4+5+6)/6 = 21/6 = 3.5
$3.50 is the break-even "fair" cost per roll.
Thanks, I did not know what the "fair die" meant.
A fair die is one that is not "loaded" or "shaved" to favor a particular side.
To determine the fair cost for each roll in the game of dots, we need to consider the expected value of a single roll. The expected value represents the average outcome we can expect from a random process like rolling a fair die.
In this game, we know that rolling the die will result in a number between 1 and 6, representing the number of dots on the top face of the die. Since the roll is fair, each of these numbers (1 to 6) has an equal chance of appearing.
To calculate the expected value, we need to multiply each possible outcome by its corresponding probability and then sum up these values. In this case, the probability of any particular number (1 to 6) appearing is 1/6 since there are six equally likely outcomes.
Expected value (E) = (1/6) * 1 + (1/6) * 2 + (1/6) * 3 + (1/6) * 4 + (1/6) * 5 + (1/6) * 6
Simplifying this equation, we get:
E = (1/6) * (1 + 2 + 3 + 4 + 5 + 6) = (1/6) * 21 = 21/6 = 3.5
Therefore, the expected value of a single roll in the game of dots is 3.5. To make the game fair, the cost for each roll should also be 3.5 dollars.
Note: The expected value represents the long-term average outcome, so in any given roll, you may win more or less than 3.5 dollars. However, over a large number of rolls, your average winnings should converge towards 3.5 dollars per roll.