This question has been bugging me for a while now and I want to know where I'm going wrong.
There are $20$ tickets in a raffle with one prize. What should each ticket cost if the prize is \$80 and the expected gain to the organizer is \$30?
Now I can get the right answer by adding \$80 and \$30 then dividing by the 20 tickets to get \$5.50 per ticket, but when I use the expected value equation such as $\frac{1}{20}(p-80) + \frac{19}{20}p = 30$ to find the price of a ticket I get a much larger value which is indeed incorrect. What am I doing wrong in my equation?
Best Answer
If $p$ is the price per ticket, then $\frac 1{20} (p−\$80)+\frac{19}{20} p$ is the expected return for selling one ticket.
You want the expected return for selling twenty tickets to equal $\$30$. Fortunately the Linearity of Expectation means this is:
$$20\times(\frac 1{20} (p−\$80)+\frac{19}{20} p)=\$30 $$
This yields $p=\$5.50$