Question

​Three students want to open an ice cream business next summer. Max thinks that demand will be high and they should purchase a large stand and acquire premium ice cream equipment. Minnie feels that demand will be low and that they should purchase a tarp and some basic hand tools to make the ice cream. Midsy is pretty sure that demand will fall in-between these two extremes, so they can get by with a shack and a used ice cream machine. You are their business consultant and estimated the following costs, revenues, and probabilities of each level of demand in the table shown below. What is the expected profit for this venture?

Demand	Total Revenue	Total Expenses	Probability
Low	$300	$50	.3
Medium	$1,200	$500	.3
High	$5,000	$850	.4

		$1,695
		$1,670
		$1,945
		$1,315

Answer 1

We have to find the expected profit.

We know that expected profit = sum of (profit for a particular state of nature*probability of occurrence of that state of nature) for all the state of nature.

Here the various state of nature, their total revenue, total cost, profit and probability of occurrence is:

State of Nature (Demand)	Total Revenue	Total Cost	Profit (Total Revenue – Total Cost)	Probability of occurrence
Low	$300	$50	$250	0.3
Medium	$1,200	$500	$700	0.3
High	$5,000	$850	$4150	0.4

Hence the profit for each demand type, profit, probability of occurrence and expected profit from that state can be denoted as:

State of Nature (Demand)	Profit	Probability of occurrence	Expected Profit (Profit * Probability of occurrence)
Low	$250	0.3	=250*0.3 = $75
Medium	$700	0.3	=700*0.3 = $210
High	$4150	0.4	=4150*0.4 = $1660

Hence the expected profit is: sum of expected profit for all demand type

= 75 + 210 + 1660

= $1945

Answer is: $1,945

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