Q$2.(20$ points$)$ The UTD cafeteria offers scones for $\backslash (\backslash$$ $1\mathrm{.}5\backslash )$ each from $8$ am to $3$ pm $.$ The scones are ordered from their supplier at the beginning of each day and are delivered before the store opens. The supplier charges $75$ cents per scone. If at $3$ pm $,$ some scones are left unsold, the cafeteria sells the remaining scones for $50$ cents each. Assume that all leftover scones are always sold by $4$ pm when the cafeteria closes. If a customer asks for a scone before $3$ pm but the cafeteria has run out, the customer always buys a bag of chips for $\backslash (\backslash$$ $1\backslash )$ instead $($after $3$ pm $,$ the customer does not buy anything instead$).$ Assume the chips are always in stock, and they are purchased from the same supplier for $40$ cents each. Demand for scones before $3$ pm at the cafeteria is variable but can only take values between $40$ and $50,$ with probabilities given in the following table.

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a$.$ Yesterday the cafeteria ordered $45$ scones, and $49$ customers came wanting to buy a scone between $8$ am and $3$ pm $.$ What was their total profit $($including$,$ if any, the profit on chips sold instead of scones$)?(6$ points$)$

b$.$ How much is the underage cost $\backslash (\backslash$mathrm$\{$C$\}\_\{\backslash$mathrm$\{$u$\}\}\backslash )?(3$ points$)$

c$.$ How much is the overage cost $\backslash (\backslash$mathrm$\{$C$\}\_\{\backslash$mathrm$\{$o$\}\}\backslash )?(3$ points$)$

d$.$ What is the optimal number of scones to order at the start of each day? $(4$ points$)$

e$.$ Given the order quantity in part d$,$ what is the probability that the cafeteria cannot satisfy all the demand for scones before $3$ pm $?(\backslash (\backslash$mathbf$\{4\}\backslash )$ points$)$