Question

..A game is played by rolling a single, six-sided die.

If the result of the roll is a 3, the player wins.
If the result of the roll is not a 3, the player loses $11.
In order for this to be a fair game, when the player wins, how much should
they win?

Answer 1

To find out how much the player should win when they roll a 3, we need to calculate the expected value of the game. The expected value tells us how much the player can expect to win or lose on average over many rolls of the die.

The probability of rolling a 3 on a single, six-sided die is 1/6. The probability of not rolling a 3 is 5/6. So the expected value of the game is:

E = (1/6) * x + (5/6) * (-11)

where x is the amount the player wins when they roll a 3. We want the expected value to be 0, since we want the game to be fair. So we can solve for x:

0 = (1/6) * x + (5/6) * (-11)
0 = (1/6) * x - 55/6
55/6 = (1/6) * x
x = 55

So the player should win $55 when they roll a 3 in order for the game to be fair.

Answer 2

The player should win $55 when they win in order for the game to be fair.

Step-by-step explanation:

Theory:

In order for the game to be fair, the expected value of the game should be $0.

Calculation:

Let's calculate the expected value of the game:

Probability of rolling a 3 (winning): 1/6

Expected win: Probability of winning * Amount won = (1/6) * X

Probability of not rolling a 3 (losing): 5/6

Expected loss: Probability of losing * Amount lost = (5/6) * (-11)

Setting the expected value to $0 and solving for X:

(1/6) * X + (5/6) * (-11) = 0

X/6 - 55/6 = 0

X/6 = 55/6

X = 55

Answer:

In order for the game to be fair, the player should win $55 when they win.