Question

A factory has produced a total of 32,000 cellphone cases. There are 3 scenarios for their demand and another 3 for their unit sale price as shown below. Remaining (unsold) units have to be heavily discounted making a profit of only $2.30 each. Determine the probability that they will make a total profit larger than $300,000.

Units sold #	Probability for units sold	Profit $/unit	Probability for profit $/unit
10,000	0.20	13.50	0.15
15,000	0.45	19.50	0.55
20,000	0.35	28.50	0.30

Answer 1

Drawing a matrix for all situations and calculating profits in each case we have

Units sold in Rows and Profit $/unit in Column and we calculate profit in each situation. (For example in the first cell, the 10,000 units sold will be at a profit of 13.50 per unit and the remaining 22,000 cases produced will have to be sold for 2.30$ profit per unit for a total profit of 13.5*10000 + 2.3*22000 (With a probability of 0.20*0.15 which is the probability that there will be 10,000 units sold at 13.50 profit per unit)

	13.5	19.5	28.5
10000	13.5*10000 + 2.3*22000 (With probability 0.20*0.15)	19.5*10000 + 2.3*22000 (With probability 0.20*0.55)	28.5*10000 + 2.3*22000 (With probability 0.20*0.30)
15000	13.5*15000 + 2.3*17000 (With probability 0.45*0.15)	19.5*15000 + 2.3*17000 (With probability 0.45*0.55)	28.5*15000 + 2.3*17000 (With probability 0.45*0.30)
20000	13.5*20000 + 2.3*12000 (With probability 0.35*0.15)	19.5*20000 + 2.3*12000 (With probability 0.35*0.55)	28.5*20000 + 2.3*12000 (With probability 0.35*0.30)

	13.5	19.5	28.5
10000	185,600$ (Probability 0.03)	245,600 $(With probability 0.11)	335,600 (With probability 0.06)
15000	241,600$ (With probability 0.0675)	331,600$ (With probability 0.2475)	466,600$ (With probability 0.135)
20000	297,600$ (With probability 0.0525)	417,600$ (With probability 0.1925)	597,600 (With probability 0.105)

Therefore the Probability that the profit is larger than 300,000$ will be the sum of the probabilities in the cells where the profit in the cell is larger than 300,000 = 0.06 + 0.2475 + 0.135 + 0.1925 + 0.105 = 0.74

Therefore the probability that they will make a total profit larger than 300,000$ is 74% or 0.74.

Hope it helps. Do ask for any clarifications if required.