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 payoff will be
(1/6)(1+2+3+4+5+6) = 21/6 of the bet.
The breakeven "fair" price for a roll should be $3.50.
Since the "house" will want to retain an advantage, they might charge $4 per roll.
To determine the cost for each roll in order for the game to be considered fair, we need to calculate the expected value.
The expected value measures the average amount we expect to win (or lose) for each roll. In this case, the payout for each roll is determined by the number of dots on the top face of the die.
A fair game means that the expected value is zero. In other words, on average, we neither win nor lose money when playing the game.
Here's how we can calculate the expected value:
1. Determine the probability of getting each number on the die:
- The die is fair, so each number from 1 to 6 has an equal probability of appearing, which is 1/6.
2. Calculate the expected value for each number:
- For each dot showing on the top face of the die, we receive $1.
- So, for each roll, the expected value is the probability multiplied by the payout:
- E(1) = (1/6) * $1 = $1/6
- E(2) = (1/6) * $1 = $1/6
- E(3) = (1/6) * $1 = $1/6
- E(4) = (1/6) * $1 = $1/6
- E(5) = (1/6) * $1 = $1/6
- E(6) = (1/6) * $1 = $1/6
3. Sum up the expected values:
- We add up the expected values for each possible outcome:
- (1/6) * $1/6 + (1/6) * $1/6 + (1/6) * $1/6 + (1/6) * $1/6 + (1/6) * $1/6 + (1/6) * $1/6
- = $1/6 + $1/6 + $1/6 + $1/6 + $1/6 + $1/6
- = $6/6
Since the sum of the expected values is $6/6, which equals $1, the cost should be set at $1 per roll for the game to be considered fair.