Question

At a raffle, 200 tickets are sold for $5 apiece. Of those tickets, 10 will win you a prize of $20, 4 will winyou a prize of $15, and 1 will win you a prize of $50. Find the expected value of buying a ticket.

Answer 1

Step 1. The information that we have is:

• 200 tickets are sold at a price of $5

,

• 10 will win a prize of $20

,

• 4 will win a prize of $15

,

• 1 will win a prize of $50

Required: Find the expected value of buying a ticket.

Step 2. We will find the value of the ticket for each case and its probability.

For those 10 people who bought a $5 ticket and won a $20 prize, the value of their ticket was:

`20-5=15`

And the probability of being one of those 10 people for whom the ticket is worth $15 is:

`P(15)=(10)/(200)`

Step 3. For the 4 people who will win a $15 prize, the value of their ticket was:

`15-5=10`

We subtract 5 because of the initial ticket they bought.

And the probability of being one of those people for whom the ticket is worth $10 is:

`P(10)=(4)/(200)`

Step 4. One will win a prize of $50, for that person, the value of their ticket is:

`50-5=45`

And the probability of being the person for whom the ticket is worth $45 is:

`P(45)=(1)/(200)`

Step 5. For the rest of the people (the rest is 185 people) who do not win any prize, the value of their ticket is a negative 5 because they spent $5 on the ticket but do not win any prize.

The probability of being one of those people for whom the ticket value is -$5 is:

`P(-5)=(185)/(200)`

Step 6. To summarize, we can make a table where x is the value of the ticket and P(x) is the probability:

Step 7. The expected value is found using the formula:

`E=x_1P_1+x_2P_2+x_3P_3+...`

Where each term represents the multiplication of one x value and its probability.

The result is:

`E=15((10)/(200))+10((4)/(200))+45((1)/(200))+(-5)((185)/(200))`

Making the operations:

`\begin{gathered} E=15(0.05)+10(0.02)+45(0.005)+(-5)(0.925) \\ E=-3.45 \end{gathered}`

The expected value is -$3.45

Answer:

-3.45