This problem has a young Billy and an older and smarter Billy.

$($a$)$ Every morning, rain or shine, young Billy Gates can be found at the entrance to the metro, hawking

copies of $$The Morningstar.$$ Demand for newspapers varies from day to day, but Billy$$s regular

early morning haul yields him $200$ copies. He purchases these copies for $$1$ per paper, and sells

them for $$1\mathrm{.}50$ apiece. Billy goes home at the end of the morning, or earlier if he sells out. He

can return unsold papers to the distributor for $$0\mathrm{.}50$ apiece. From experience, Billy knows that

demand for newspapers on any given morning is uniformly distributed between $150$ and $250,$ where

each of the possible values $150,151,...,250$ is equally likely. What is the expected value of Billy$$s

net earnings on any given day?