Answer:
The expected winnings for a person buying 1 ticket is -0.2.
Step-by-step explanation:
Given : A raffle offers a first prize of $1000, 2 second prizes of $300, and 20 third prizes of $10 each. If 20000 tickets are sold at 25 cents each, find the expected winnings for a person buying 1 ticket.
To find : What are the expected winnings?
Solution :
There are one first prize, 2 second prize and 20 third prizes.
Probability of getting first prize is ![\frac{1}{20000}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B20000%7D)
Probability of getting second prize is ![\frac{2}{20000}](https://tex.z-dn.net/?f=%5Cfrac%7B2%7D%7B20000%7D)
Probability of getting third prize is ![\frac{20}{20000}](https://tex.z-dn.net/?f=%5Cfrac%7B20%7D%7B20000%7D)
A raffle offers a first prize of $1000, 2 second prizes of $300, and 20 third prizes of $10 each.
So, The value of prizes is
![\frac{1}{20000}\times 1000+\frac{2}{20000}\times 300+\frac{20}{20000}\times 10](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B20000%7D%5Ctimes%201000%2B%5Cfrac%7B2%7D%7B20000%7D%5Ctimes%20300%2B%5Cfrac%7B20%7D%7B20000%7D%5Ctimes%2010)
If 20000 tickets are sold at 25 cents each i.e. $0.25.
Remaining tickets = 20000-1-2-20=19977
Probability of getting remaining tickets is ![\frac{19977}{20000}](https://tex.z-dn.net/?f=%5Cfrac%7B19977%7D%7B20000%7D)
The expected value is
![E=\frac{1}{20000}\times 1000+\frac{2}{20000}\times 300+\frac{20}{20000}\times 10-\frac{19977}{20000}\times 0.25](https://tex.z-dn.net/?f=E%3D%5Cfrac%7B1%7D%7B20000%7D%5Ctimes%201000%2B%5Cfrac%7B2%7D%7B20000%7D%5Ctimes%20300%2B%5Cfrac%7B20%7D%7B20000%7D%5Ctimes%2010-%5Cfrac%7B19977%7D%7B20000%7D%5Ctimes%200.25)
![E=\frac{1000+600+200-4994.25}{20000}](https://tex.z-dn.net/?f=E%3D%5Cfrac%7B1000%2B600%2B200-4994.25%7D%7B20000%7D)
![E=\frac{-3194.25}{20000}](https://tex.z-dn.net/?f=E%3D%5Cfrac%7B-3194.25%7D%7B20000%7D)
![E=-0.159](https://tex.z-dn.net/?f=E%3D-0.159)
Therefore, The expected winnings for a person buying 1 ticket is -0.2.