$(12$ points$)$ A busy bakery in town experiences a constant flow of customers during peak

hours. The front door is observed to admit $3$ customers every $2$ minutes. For a stable

system over a longer period, the number of customers entering should equal the

number leaving.

a$.(6$ points$)$ What is the throughput of the bakery?

b$.(6$ points$)$ How long, on average, does a customer spend in the bakery $($T$,$

expressed in units of time$)?$

$(12$ points$)$ You are a senior analyst at Triton Investment Group $($TIG$)$ and are

considering an investment in a San Diego$-$based tech startup named Health$-$InnoTech.

You have been tasked with assessing the risks associated with TIG investing in Health$-$

InnoTech.

If TIG opts to invest, there's a $30\%$ chance that Health$-$InnoTech will thrive post$-$

investment, and a $70\%$ chance of failure. A success would net TIG $$150,000,$ but

a failure would result in no return.

If TIG decides against investing in Health$-$InnoTech, the funds will be directed to

either bonds or real estate. The gain from bonds is $$25,000$ if the financial

market is favorable, and $$5,000$ if not. The probability of a favorable bond

market is $50\%,$ and $50\%$ for an unfavorable one.

a$.(4$ points$)$ Draw a decision tree that shows all of the above outcomes.

b$.(4$ points$)$ Calculate the expected monetary value for both scenarios:

$($i$)$ TIG invests in Health$-$InnoTech, and

$($ii$)$ TIG does not invest in Health$-$InnoTech.

c$.(4$ points$)$ Based on the expected values, should TIG invest in Health$-$

InnoTech or not? Explain your recommendation.

$(11$ points$)$ An upcoming train service is nearly at capacity with only $10$ seats

remaining. The standard tickets are priced at $$150$ each, while discounted tickets go

for $$50$ each. The demand for standard fare passengers is normally distributed with an

average of $15$ and a standard deviation of $5.$

a$.(8$ point$)$ How many seats should the train company reserve for the standard fare

class?

b$.(3$ point$)$ Based on your answer above, how much total revenue will be earned

from this train service?