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 consdiered a fair game?

How many outcomes are there when we throw a 6-faced die?

How many dots are there for each outcome?
Add up the dots for each outcome and divide by the number of outcomes would give you an "average" payout, which you could use for the fair cost of each roll.

In order to determine the cost that should be set for each roll in order for the game to be considered fair, we need to consider the probabilities and payoffs involved.

In this game, we have a fair die, meaning that each face (1, 2, 3, 4, 5, and 6) has an equal chance of appearing. The objective is to find the expected value, which represents the average amount of money won or lost per roll.

To calculate the expected value, we multiply each possible outcome by its corresponding probability and sum them up. In this case, the possibilities are the numbers of dots showing on the die's top face (1, 2, 3, 4, 5, and 6), and the payoffs are $1 for each dot showing.

The probabilities for each outcome are equal since the die is fair, with each face having a 1/6 chance of appearing. So, the expected value is:

(1/6) * $1 + (1/6) * $2 + (1/6) * $3 + (1/6) * $4 + (1/6) * $5 + (1/6) * $6

Simplifying, we get:

$1/6 + $2/6 + $3/6 + $4/6 + $5/6 + $6/6
$21/6

So, the expected value is $21/6 = $3.50.

For the game to be considered fair, the cost for each roll should be equal to the expected value of $3.50. Therefore, the cost to play each round of the game of dots should be set to $3.50.