Five thousand raffle tickets are to be sold at $10 each to benefit a local community group. The prizes, the number of each prize to be given away, and the dollar value of winnings for each prize are as follows:
Prize Number to Be
Given Away Dollar Value
Automobile 1 $20,000
Entertainment center 2 3,000 each
DVD recorder 5 400 each
Gift certificate 50 20 each
If you buy one ticket, calculate your expected winnings. (Form the probability distribution of x = your dollar winnings, and remember to subtract the cost of your ticket.)(Round your answer to 2 decimal places. Negative amounts should be indicated by a minus sign. Omit the "$" sign in your response.)
To calculate your expected winnings, we need to determine the probability distribution of your dollar winnings.
Let's first calculate the probability of winning each prize:
1) Automobile: There is only 1 automobile prize, so the probability of winning it is 1/5000.
2) Entertainment center: There are 2 entertainment center prizes, so the probability of winning one is 2/5000. However, since there are two prizes, we should multiply this probability by 2 to get the total probability of winning an entertainment center prize.
3) DVD recorder: There are 5 DVD recorder prizes, so the probability of winning one is 5/5000. Again, since there are five prizes, we multiply this probability by 5 to get the total probability of winning a DVD recorder prize.
4) Gift certificate: There are 50 gift certificate prizes, so the probability of winning one is 50/5000. Once more, we multiply this probability by 50 to get the total probability of winning a gift certificate prize.
Next, we need to calculate the expected winnings for each prize:
1) Automobile: The dollar value of the prize is $20,000. However, since we bought a ticket for $10, we need to subtract this cost from the prize value. Therefore, the expected winnings for the automobile prize are: $20,000 - $10 = $19,990.
2) Entertainment center: The dollar value of each entertainment center prize is $3,000. Subtracting the ticket cost, the expected winnings for each prize will be $3,000 - $10 = $2,990. Since there are two prizes, we multiply this value by 2 to get the total expected winnings for entertainment centers: $2,990 * 2 = $5,980.
3) DVD recorder: The dollar value of each DVD recorder prize is $400. After subtracting the ticket cost, the expected winnings for each prize is $400 - $10 = $390. Multiplying this value by 5 (the number of prizes), we get the total expected winnings for DVD recorders: $390 * 5 = $1,950.
4) Gift certificate: The dollar value of each gift certificate prize is $20. Subtracting the ticket cost, the expected winnings for each prize is $20 - $10 = $10. Multiplying this value by 50 (the number of prizes), we get the total expected winnings for gift certificates: $10 * 50 = $500.
Finally, we can calculate the overall expected winnings by summing up the expected winnings for each prize:
Overall expected winnings = (probability of winning automobile prize * expected winnings for automobile prize) + (probability of winning entertainment center prize * expected winnings for entertainment center prize) + (probability of winning DVD recorder prize * expected winnings for DVD recorder prize) + (probability of winning gift certificate prize * expected winnings for gift certificate prize)
= (1/5000 * $19,990) + (2/5000 * $5,980) + (5/5000 * $1,950) + (50/5000 * $500)
Calculating this expression, we find that the overall expected winnings is $10.48.
Therefore, if you buy one ticket, your expected winnings are $10.48.