Problem $20-8($Algo$)$

Lakeside Bakery bakes fresh pies every morning. The daily demand for its apple pies is a random variable with $($discrete$)$ distribution, based on past experience, given by

Demand $61319232731$

Probability $4\%16\%32\%25\%16\%7\%$

Each apple pie costs the bakery $$11\mathrm{.}50$ to make and is sold for $$29.$ Unsold apple pies at the end of the day are purchased by a nearby soup kitchen for $79$ cents each. Assume no goodwill cost.

a$.$ If the company decided to bake $19$ apple pies each day, what would be their expected profit? $($Do not round intermediate calculations. Round your answer to $2$ decimal places.$)$

b$.$ Based on the demand distribution above, how many apple pies should the company bake each day to maximize its expected profit?