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?
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$\begingroup$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 $p=\$5.50$
$\endgroup$ 3 $\begingroup$Your mistake (in your own calculation) is that you assume all tickets are sold and the winner is the last person who buys the last ticket. This is not what always happens. Any of the tickets can win (including the first one!) and it is likely that some of them are not sold at all.
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