Question

#1: exercise set 10-1 #36

#2: if a game is not fair, explain in your own words how the winning payout amount can be calculated to make sure that it is fair?

Explanation:

Define a fair game

A fair game has an expected value of zero.

Set up expected value

Let \(P_w\) be the probability of winning.
Let \(P_l\) be the probability of losing.
Let \(W\) be the net winning payout.
Let \(C\) be the cost to play.

Formulate the equation

The expected value equation is:

$$E(X) = W \cdot P_w - C \cdot P_l = 0$$

Solve for winning payout

Rearranging the equation to solve for \(W\):

$$W \cdot P_w = C \cdot P_l$$

$$W = \frac{C \cdot P_l}{P_w}$$

Interpret the result

The fair payout equals cost times loss probability divided by win probability.

Answer:

To make a game fair, the expected value of the game must be equal to zero.

If it costs a certain amount to play, the winning payout amount can be calculated using the probabilities of winning and losing:

1. Identify the cost to play the game and the probability of losing. Multiply these two values to find the expected loss.
2. Divide this expected loss by the probability of winning.

This calculated value is the net winning payout required to make the game fair:

$$\text{Winning Payout} = \frac{\text{Cost to Play} \times \text{Probability of Losing}}{\text{Probability of Winning}}$$