To calculate the expected value for a person who buys a raffle ticket, we need to consider the potential outcomes and their probabilities.
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Winning the Raffle:
- If a person wins, they receive $1400.
- The probability of winning is \( \frac{1}{753} \) (since there is 1 winning ticket out of 753 sold).
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Not Winning the Raffle:
- If a person does not win, they receive nothing.
- The probability of not winning is \( \frac{752}{753} \) (since there are 752 losing tickets).
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Net Money Calculation:
- If the person wins, their net gain will be \( 1400 - 13 = 1387 \) (the prize minus the cost of the ticket).
- If the person loses, their net gain will be \( 0 - 13 = -13 \) (losing the cost of the ticket).
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Expected Value Calculation: The expected value (EV) can be calculated as follows:
\[ EV = (P(\text{win}) \times \text{net gain if win}) + (P(\text{lose}) \times \text{net gain if lose}) \]
Substituting in the values we derived:
\[ EV = \left( \frac{1}{753} \times 1387 \right) + \left( \frac{752}{753} \times -13 \right) \]
Calculating each term separately:
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For the winning outcome: \[ \frac{1}{753} \times 1387 \approx 1.842 \]
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For the losing outcome: \[ \frac{752}{753} \times -13 \approx -12.934 \]
Now, adding these two results for the overall expected value:
\[ EV \approx 1.842 - 12.934 \approx -11.092 \]
Thus, the expected value for someone who buys a ticket is approximately -11.09 dollars.
This means that, on average, a ticket buyer can expect to lose about $11.09 from the $13 ticket price.
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