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?
The average number of dots will be...
1+2+3+4+5+6 all divided by 6
Right?
To determine the fair cost for each roll in the game of dots, we need to consider the expected value. The expected value is the average amount of money one can expect to win or lose per roll.
In this game, there are six possible outcomes when rolling a fair die: 1, 2, 3, 4, 5, or 6 dots. Each outcome has an equal probability of occurring, which is 1/6 since there are six equally likely outcomes.
To find the expected value, we calculate the average amount won per roll by multiplying the probability of each outcome by the amount won for that outcome, and then summing them up.
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 + $2 + $3 + $4 + $5 + $6) / 6
Expected Value = $21 / 6
Expected Value = $3.50
This means that the average amount won per roll is $3.50.
For the game to be fair, the cost for each roll should be set at $3.50. This way, on average, players would neither win nor lose money over many rolls.
Note: It is worth mentioning that in practice, game organizers might round the cost up or down to make it more convenient when dealing with small currency denominations, but the fair cost concept would still remain the same.