$($Excel$)$ A manager is trying to decide whether to buy one machine or two. If only one is purchased and demand proves to be excessive $($high$),$ the second machine can be purchased later $($a second machine will not be purchased if demand is low$).$ Some sales will be lost, however because the manager won't have that second machine right away. Now, the cost per machine is lower if both machines are purchased at the same time. The probability of low demand is estimated to be $0\mathrm{.}20($therefore the probability of high demand is $8).$ The profits the manager receives when purchasing the two machines together is $$90,000$ if demand is low and $$180,000$ if demand is high. If one machine is purchased and demand is low, profit is $$110,000.$ If demand is high, the manager has three options: $1)$ do nothing $($profit $=$$$120,000),2)$ buy a second machine but at a higher price $($profit $=$$$160,000),$ or $3)$ periodically leasing a second machine as needed which will keep costs lower $($profit $=140,000)).$ The manager currently isn't favoring any one of these three options more than another $($so the probability of each can be thought of as $\frac{1}{3}).$

a$.$ Draw a decision tree for this problem.

b$.$ How many machines should the company buy initially? What is the expected payoff for this alternative?

c$.$ What is the expected payout from buying $1?$ What about buying $2?$

Thanks in advance!