50% chanceLoss of 25 $50% chanceWin of X $ X should be at least          $ to make the lottery acceptable.

1. Break Down the Problem

We need to find the value of $X$ that makes participating in this lottery acceptable, considering the probabilities and outcomes involved.

2. Relevant Concepts

To find the acceptable value of $X$, we can use the concept of expected value (EV). The EV of a gamble is calculated as follows:

$EV = (P_{Win} \times Win) + (P_{Loss} \times Loss)$

Where:

• $P_{Win} = 0.5$ (50% chance to win)
• $Win = X$ (amount won)
• $P_{Loss} = 0.5$ (50% chance to lose)
• $Loss = -25$ (amount lost)

We want the expected value to be at least 0 (breaking even), so we set up the equation:

$0 = (0.5 \times X) + (0.5 \times -25)$

3. Analysis and Detail

Now let's solve the equation step by step:

1. Set up the equation: $0 = 0.5X - 12.5$

2. Rearranging terms: $0.5X = 12.5$

3. Solving for $X$: $X = \frac{12.5}{0.5} = 25$

4. Verify and Summarize

The expected value calculation confirms that if $X = 25$, you neither lose nor gain; thus, $X$ should be at least $25 for the lottery to be acceptable.

Final Answer

The value of $X$ should be at least $25 to make the lottery acceptable.