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:
[tex]20-5=15[/tex]And the probability of being one of those 10 people for whom the ticket is worth $15 is:
[tex]P(15)=\frac{10}{200}[/tex]Step 3. For the 4 people who will win a $15 prize, the value of their ticket was:
[tex]15-5=10[/tex]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:
[tex]P(10)=\frac{4}{200}[/tex]Step 4. One will win a prize of $50, for that person, the value of their ticket is:
[tex]50-5=45[/tex]And the probability of being the person for whom the ticket is worth $45 is:
[tex]P(45)=\frac{1}{200}[/tex]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:
[tex]P(-5)=\frac{185}{200}[/tex]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:
[tex]E=x_1P_1+x_2P_2+x_3P_3+...[/tex]Where each term represents the multiplication of one x value and its probability.
The result is:
[tex]E=15(\frac{10}{200})+10(\frac{4}{200})+45(\frac{1}{200})+(-5)(\frac{185}{200})[/tex]Making the operations:
[tex]\begin{gathered} E=15(0.05)+10(0.02)+45(0.005)+(-5)(0.925) \\ E=-3.45 \end{gathered}[/tex]The expected value is -$3.45
Answer:
-3.45
