Question 1186121: Tickets for a raffle cost $12. There were 713 tickets sold. One ticket will be randomly selected as the winner, and that person wins $1800 and also the person is given back the cost of the ticket. For someone who buys a ticket, what is the Expected Value (the mean of the distribution)?
Found 3 solutions by CPhill, ikleyn, greenestamps:
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to calculate the expected value of a raffle ticket:
**1. Calculate the probability of winning:**
* There is 1 winning ticket out of 713 total tickets.
* Probability of winning = 1/713
**2. Calculate the net winnings if you win:**
* You win $1800, and you get your $12 ticket cost back.
* Net winnings = $1800 + $12 = $1812
**3. Calculate the net loss if you lose:**
* You lose the cost of the ticket.
* Net loss = -$12
**4. Calculate the expected value:**
Expected Value = (Probability of Winning * Net Winnings) + (Probability of Losing * Net Loss)
Expected Value = (1/713 * $1812) + (712/713 * -$12)
Expected Value ≈ $2.54 + (-$11.96)
Expected Value ≈ -$9.42
**Therefore, the expected value of the raffle ticket is approximately -$9.42.** This means that on average, a person buying a ticket can expect to lose about $9.42.
Answer by ikleyn(52790)  (Show Source):
You can put this solution on YOUR website! .
Tickets for a raffle cost $12. There were 713 tickets sold. One ticket will be randomly selected
as the winner, and that person wins $1800 and also the person is given back the cost of the ticket.
For someone who buys a ticket, what is the Expected Value (the mean of the distribution)?
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I do not agree with the solution in the post by @CPhill.
He calculates the net winning of the winner as
Net winnings = $1800 + $12 = $1812.
@CPhill adds $12 to the winning, but forgets to subtract $12 for buying the ticket.
The correct way to calculate and the correct value of the net winning is, of course,
- $12 + $1800 + $12 = $1800.
Accordingly, all other numbers in the solution should be changed.
After that explanation, see my correct solution below.
S O L U T I O N
Here's how to calculate the expected value of a raffle ticket:
**1. Calculate the probability of winning:**
* There is 1 winning ticket out of 713 total tickets.
* Probability of winning = 1/713
**2. Calculate the net winnings if you win:**
* You spent @12 to buy a ticket, win $1800, and you get your $12 ticket cost back. <<<---=== my corrections start from this point
* Net winnings = - $12 + $1800 + $12 = $1800
**3. Calculate the net loss if you lose:**
* You lose the cost of the ticket.
* Net loss = -$12
**4. Calculate the expected value:**
Expected Value = (Probability of Winning * Net Winnings) + (Probability of Losing * Net Loss)
Expected Value = (1/713 * $1800) + (712/713 * (-$12))
Expected Value ≈ -$-9.458625526 <<<---=== notice the difference with the value by -$9.42 in the solution by @CPhill
<<<---=== also notice that I do not round it to cents, since this value is the statistical quantity
**Therefore, the expected value of the raffle ticket is approximately -$9.46.** This means that on average, a person buying a ticket can expect to lose about $9.46.
Solved correctly.
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The difference may seem not to be significant, but in this problem not the magnitude of the difference
plays a role, but the correctness of the approach, developing right logic and the correctness of steps.
For the solution by @CPhill, strict examiner would give a solid  and would send him for retraining.
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
Tutor @ikleyn solves the problem using the formal definition of expected value; that is probably what the student was expected to do.
For simple problems like this, the expected value can be determined less formally with less work.
The total cost of the 713 tickets sold was 713($12) = $8556.
The total payout was $1800+$12 = $1812 (the person who won the prize also got back the cost of his ticket).
The net loss to the players was $8556-1812 = $6744.
The average loss for the 713 tickets sold was $6744/713 = $9.4586.... or about $9.46
ANSWER: The expected value for each ticket sold was -$9.46
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