Question $2(1\mathrm{.}5$ point$)$

Bill's Bakery bakes fresh bagels each morning. The daily demand for bagels is a random variable with a distribution estimated from prior experience given by

$\backslash$table$[[$Number sold in a day $($dozen$),$Probability$],[0,0\mathrm{.}05],[1,0\mathrm{.}10],[2,0\mathrm{.}10],[3,0\mathrm{.}20],[4,0\mathrm{.}25],[5,0\mathrm{.}15],[6,0\mathrm{.}10],[7,0\mathrm{.}05]]$

The bagels cost Billy's $8$ cents to make, and they are sold for $37$ cents each. Bagels unsold at the end of the day are purchased by Fisl & Loaves, a nearby charity soup kitchen for $3$ cents each. Based on the given discrete distribution, how many dozens of bagels should Billy's bake at the start of each day?

$($Show all details, e$.$g$.,$ the overage, and underage costs, and the critical fractile in your answer$)$