"Suppose you have an opportunity to buy a ticket to a concert that includes your favorite band/performer. You have a choice about how much to pay for the ticket. You have 200 dollars in your wallet. A ticket normally costs 100. Ticketmeister has tickets for sale at a discounted rate of 56 dollars for the same quality seats. HOWEVER, in order to purchase the last discounted ticket, you will have to park illegally. The cost of a parking ticket is $80.
Ceteris paribus, if you only care about saving money on the cost of the concert ticket, you should park illegally only if the probability of getting caught is below ____%. (Fill in the blank)"
I need to know what the probability of getting caught as a function of the concert ticket price, but I have absolutely no idea where to start with this problem!
$C_p=56+80=136$ is the cost if caught.
$C_{1-p}=56+ 0=56$ is the cost if not.
The expected cost is then $$\Bbb E(C)=pC_p+(1-p)C_{1-p}=136p+56(1-p)=80p+56$$ and we have $$\Bbb E(C)=80p+56\le 100\quad\text{if}\quad p\le\frac{44}{80}=0.55=55\%$$