To determine if a game is fair, we calculate the expected value (EV) of the outcomes. If the expected value is 0, the game is fair.

The formula for expected value is:

$EV = \sum (x \cdot p(x))$

Where:

• $x$ = value of each outcome
• $p(x)$ = probability of each outcome

If probabilities are not specified, it is assumed that all outcomes are equally likely.

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(a) Outcomes: $4, 2, -1, 5$

If there are 4 outcomes, and each is equally likely, the probability of each outcome is $\frac{1}{4}$.

$EV = \frac{1}{4}(4) + \frac{1}{4}(2) + \frac{1}{4}(-1) + \frac{1}{4}(5)$

$EV = \frac{4 + 2 - 1 + 5}{4} = \frac{10}{4} = 2.5$

Since the expected value is $2.5$, this is not a fair game.

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(b) Outcomes: $5, -10, -1, 2$

Each outcome is equally likely, so the probability of each is $\frac{1}{4}$.

$EV = \frac{1}{4}(5) + \frac{1}{4}(-10) + \frac{1}{4}(-1) + \frac{1}{4}(2)$

$EV = \frac{5 - 10 - 1 + 2}{4} = \frac{-4}{4} = -1$

Since the expected value is $-1$, this is not a fair game.

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(c) Outcomes: $5, 9, 2, 4$

Each outcome is equally likely, so the probability of each is $\frac{1}{4}$.

$EV = \frac{1}{4}(5) + \frac{1}{4}(9) + \frac{1}{4}(2) + \frac{1}{4}(4)$

$EV = \frac{5 + 9 + 2 + 4}{4} = \frac{20}{4} = 5$

Since the expected value is $5$, this is not a fair game.

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Summary of Results:

1. (a) Not a fair game.
2. (b) Not a fair game.
3. (c) Not a fair game.

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Would you like a deeper breakdown or more related examples?

Expanding Questions:

1. What is the significance of the expected value in determining fairness?
2. How does unequal probability affect the fairness of a game?
3. Can you provide an example of a fair game with specified probabilities?
4. What if additional outcomes were added—how would it impact the fairness?
5. How can this analysis apply to real-world scenarios, such as gambling or business decisions?

Tip: Always verify the probabilities sum to 1 when dealing with unequal likelihoods in games!