Famous Albert prides himself on being the Cookie King of the West. Small, freshly baked cookies are the specialty of his shop. Famous Albert has asked for help to determine the number of cookies he should make each day. From an analysis of past demand, he estimates demand for cookies as

$\backslash$table$[[$DEMAND$,\backslash$table$[[$PROBABILITY OF$],[$DEMAND$]]],[1,800$ dozen,$0\mathrm{.}03],[2,000,0\mathrm{.}07],[2,200,0\mathrm{.}20],[2,400,0\mathrm{.}27],[2,600,0\mathrm{.}13],[2,800,0\mathrm{.}03],[3,000,0\mathrm{.}27]]$

Each dozen sells for $$0\mathrm{.}70$ and costs $$0\mathrm{.}51,$ which includes handling and transportation. Cookies that are not sold the end the are reduced to $$0\mathrm{.}29$ and sold the following day as day$-$old merchandise.

a$.$ Compute the expected profit or loss for each cookie making decision quantity. $($Round your answer to the nearest whole number. Enter expected losses with a negative sign.$)$

$\backslash$table$[[\backslash$table$[[$Cookies Baked$],[($Dozen$)]],\backslash$table$[[$Probability of$],[$Demand$]],\backslash$table$[[$Expected$],[$Profit$/$Loss$]]],[1,800,0\mathrm{.}03,],[2,000,0\mathrm{.}07,],[2,200,0\mathrm{.}20,],[2,400,0\mathrm{.}27,],[2,600,0\mathrm{.}13,],[2,800,0\mathrm{.}03,],[3,000,0\mathrm{.}27,]]$

b$.$ Based on your answers to part a$.,$ what is the optimal number of cookies to make?

Optimal number of cookies

dozen