Every year in early October Steven King buys pumpkins of one size from a farmer in Maine and then hires an artist to carve bewitching faces in them. He then tries to sell them at his produce stand in a public market in Boston.$$The farmer charges Steven $$1\mathrm{.}50$ per pumpkin and the artist is paid $$2\mathrm{.}00$ per carved pumpkin. Steven sells a carved pumpkin for $$15\mathrm{.}00.$ Any pumpkins not sold by $5$:$00$ p$.$m$.$ on Halloween are donated to Steven's favorite children's hospital. Steven pays the artist $$1\mathrm{.}00$ per pumpkin to rush the pumpkins to the hospital for the youngsters to enjoy.$$Steven estimates the demand for his pumpkins this season to be approximately normally distributed with mean of $100$ units and a standard deviation of $40$ units.

$($a$).$ How many pumpkins should Steven have available for sale? $(30$ points$)$

$($b$).$ How likely will Steven experience a stock$-$out if he orders the amount you recommended in $($a$)?(10$ points$)$