Theory of Expectation :: Problems on Profits, Business : Probability Distribution

No. Problems & Solutions [Click Hide/Show to display the solutions below the question]
01. In a business venture a man can make a profit of Rs. 2,000 with a probability of 0.4 or have a loss of Rs. 1,000 with a probability of 0.6. What is his expected profit?

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02. Let "x" denote the profit that a man makes in a business. He may earn Rs. 2,800 with probability 0.5 he may lose Rs. 5,500 with a probability 0.3 and he may neither earn nor lose with a probability of 0.2. Calculate the mathematical expectation of "x".

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03. If it rains, a taxi driver can earn Rs.100 per day. If it is fair, he can lose Rs.10 per day. If the probability of rain is 0.4, what is his expectation?

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04. A company introduces a new product in the market and excepts to make a profit of Rs. 2.5 lakhs during the first year if the demand is "good", a profit of Rs. 1.5 lakhs if the demand is "moderate" and a loss of Rs. 1 lakh if the demand is "poor". Market Research Studies indicate that the probabilities for the demand to be good and moderate are 0.2 and 0.5 respectively.

Find the company's expected profit and the standard deviation of profit.

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05. A building contractor is considering one of the two contracts for new down town buildings "A" and "B". It has been estimated that a profit of Rs. 2,00,000 would be made on building "A". Bidding costs for the contractor on building "A" would be Rs. 10,000. On building "B", the estimated profit is Rs. 5,00,000 and bidding costs would be Rs. 20,000. The probability of the contract being awarded is 2/5 for building "A" and 1/5 for building "B" (Assume bidding costs are incurred only in case the contract is not obtained).

i) What is the contractor's expectation for building A?
ii) What is the contractor's expectation for building B?
iii) For which Job should the contractor bid?

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06. A consignment of machine parts is offered to two firms, "A" and "B", for Rs.75,000. The following table shows the probabilities at which the firms "A" and "B" will be able to sell the consignment at different prices:
Probabilities Prices (Rs.) at which Consignment can be sold
60,000 70,000 80,000 90,000
A: 0.40 0.30 0.20 0.10
B: 0.10 0.20 0.50 0.20
Which firms, A or B will be more inclined towards this offer?

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07. Under an employment promotion programme, it is proposed to allow sale of news papers on the buses during off-peak hours. The vender can purchase the newspaper at a special concessional rate of 25 paise per copy against the selling price of 40 paise. Any unsold copies are however, a dead loss. A vendor has estimated the following probability distribution of the number of copies demanded
Number of copies 15 16 17 18 19 20
Probability 0.04 0.19 0.33 0.26 0.11 0.07
How many copies should he order so that his expected profit will be a maximum?

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08. The proprietor of a food stall has introduced a new item of food. The cost of making it is Re. 1 per piece, and because of its novelty it could be sold for Rs. 3 per piece. It is, however, perishable and pieces remaining unsold at the end of a day are a dead loss. He expects the demand to be variable and has drawn up the folloing probability distribution expressing his estimates.
Number of pieces demand 10 11 12 13 14 15
Probability 0.07 0.10 0.23 0.38 0.12 0.10
Calculate the expected profit or loss if he manufactures 12 pieces.

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09. A firm plans to bid Rs. 300 per tonne for a contract to supply 1,000 tonnes of a metal. It has two competitors "A" and "B" and it assumes that the probability that "A" will bid less than Rs. 300 per tonne is 0.7. If the lowest bidder gests all the business and the firms bid independently, what is expected value of the contract to the firm?

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No. Problems for Practice
01. If it rains, an umbrella sales man can earn Rs. 300 per day. If it is fair he can lose Rs. 60 per day. What is his expectation if the probability of rain is 0.3?
02. Anil company estimated the net profit on a new product it is launching to be Rs. 30,00,000 during the first year if it is "successful"; Rs.10,00,000 if it is "moderately successful" and a loss of Rs. 10,00,000 if it is “unsuccessful". The firm assigns the following probabilities to first year for the product successful – 0.45, moderately successful – 0.25. What are the expected value and standard deviation of first year net profit for this product?

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