In a raffle 1000 tickets are being sold at $1.00 each. the first prize is $100, and there are 3 second prizes of $50 each. by how much does the price of a ticket exceed its expected value?
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(-1)(.999)+(99)(.001)+(49)(1/999)+(49)(1/998)+(49)(1/997)=-.75
To find the expected value of a ticket, we need to calculate the total value of the prizes and divide it by the number of tickets sold.
The value of the first prize is $100, and there are 3 second prizes of $50 each. So the total prize value is:
(1 x $100) + (3 x $50) = $100 + $150 = $250.
Since there are 1000 tickets being sold at $1.00 each, the total revenue from ticket sales is:
1000 x $1 = $1000.
Now, we can calculate the expected value per ticket by dividing the total prize value by the number of tickets sold:
$250 / 1000 = $0.25.
The expected value of a ticket is $0.25.
To calculate how much the price of a ticket exceeds its expected value, we subtract the expected value from the price of a ticket:
$1.00 - $0.25 = $0.75.
Therefore, the price of a ticket exceeds its expected value by $0.75.
To calculate the expected value of a ticket, we need to determine the probability of winning each prize and multiply it by the value of the prize. Given the information provided, let's calculate the expected value for a single ticket step-by-step:
First prize:
- The probability of winning the first prize is 1 out of 1000 since there is only one first prize.
- The value of the first prize is $100.
Second prizes:
- The probability of winning a second prize is 3 out of 1000 since there are three second prizes.
- The value of each second prize is $50.
Let's calculate the expected value for the first and second prizes:
Expected value for the first prize = (Probability of winning) * (Value of the prize)
= (1/1000) * ($100)
= $0.10
Expected value for the three second prizes = (Probability of winning) * (Value of each prize) * (Number of prizes)
= (3/1000) * ($50) * (3)
= $0.45
Now let's calculate the total expected value of a ticket:
Total expected value = Expected value for the first prize + Expected value for the second prizes
= $0.10 + $0.45
= $0.55
Since the cost of a ticket is $1.00, we can find the amount by which the price of a ticket exceeds its expected value by subtracting the expected value from the cost:
Excess price of a ticket = Cost of a ticket - Total expected value
= $1.00 - $0.55
= $0.45
Therefore, the price of a ticket exceeds its expected value by $0.45.