Question $3.(15$ Points$)$ Sweetie, a chocolate manufacturer, produces heart shaped chocolates for Valentine's Day. A$102$ makes a single ordering decision at the end of January every year for that year's Valentine's Day. The retail price for the chocolate is $ $8.$ Sweetie charges A$102$ $ $4$ for the chocolate and a stock$-$out situation at A$102$ results in $ $3.$ Sweetie can sell an unsold chocolate to a $3^$rd party for $ $1$ per unit. If demand is a random variable with discrete distribution with the following probability distribution: a$)(5$ points$)$ What is the optimal order quantity for A$102?$ b$)(10$ points$)$ What is the optimal expected profit?