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

Solution

Profit earned = [(No. of Copies Sold) × (Selling Price)]     − [(No. of Copies Bought) × (Purchase Price)] = (No. of Copies Sold) × (Rs. 0.40)     − [(No. of Copies Bought) × (Rs. 0.25)] The vendor may buy either 15 or 16 or 17 or 18 or 19 or 20 copies and at the same time the demand for the news papers may also be either 15 or 16 or 17 or 18 or 19 or 20 copies Sale realisation on sale of 15 copies ⇒ 15 × 0.40 = Rs. 6 16 copies ⇒ 16 × 0.40 = Rs. 6.40 17 copies ⇒ 17 × 0.40 = Rs. 6.80 18 copies ⇒ 18 × 0.40 = Rs. 7.20 19 copies ⇒ 19 × 0.40 = Rs. 7.60 20 copies ⇒ 20 × 0.40 = Rs. 8 Cost incurred on purchse of 15 copies ⇒ 15 × 0.25 = Rs. 3.75 16 copies ⇒ 16 × 0.25 = Rs. 4.00 17 copies ⇒ 17 × 0.25 = Rs. 4.25 18 copies ⇒ 18 × 0.25 = Rs. 4.50 19 copies ⇒ 19 × 0.25 = Rs. 4.75 20 copies ⇒ 20 × 0.25 = Rs. 5.00 Therefore, the Profit earned = Sale Realisation − Cost Incurred In this decision making problem Since the decision with regard to how many copies should be bought is in the hands of the vendor, no. of copies to be bought would form the acts. Since how many copies would be demanded is something not within the hands or control of the vendor, the no. of copies demanded would form the states of nature. Each state of nature has a certain probability of occurance. The probabilities for the states are not varying with the acts i.e. the probability for a state of nature is the same whatever may be act that the vendor chooses. Calculations for profits made under various situations No. Bought → Demanded ↓ 15 16 17 18 19 20 15 2.25 2.00 1.75 1.50 1.25 1.00 16 2.25 2.40 2.15 1.90 1.65 1.40 17 2.25 2.40 2.55 2.30 2.05 1.80 18 2.25 2.40 2.55 2.70 2.45 2.20 19 2.25 2.40 2.55 2.70 2.85 2.60 20 2.25 2.40 2.55 2.70 2.85 3.00 Calculations for expected profit under various situations Bought → 15 16 17 18 19 20 Demanded ↓ Prob Pr Exp Pr Pr Exp Pr Pr Exp Pr Pr Exp Pr Pr Exp Pr Pr Exp Pr 15 0.04 2.25 0.09 2.00 0.08 1.75 0.07 1.50 0.06 1.25 0.05 1.00 0.04 16 0.19 2.25 0.43 2.40 0.46 2.15 0.41 1.90 0.36 1.65 0.31 1.40 0.27 17 0.33 2.25 0.74 2.40 0.79 2.55 0.84 2.30 0.76 2.05 0.68 1.80 0.59 18 0.26 2.25 0.59 2.40 0.62 2.55 0.66 2.70 0.70 2.45 0.64 2.20 0.57 19 0.11 2.25 0.25 2.40 0.26 2.55 0.28 2.70 0.30 2.85 0.31 2.60 0.29 20 0.07 2.25 0.16 2.40 0.17 2.55 0.18 2.70 0.19 2.70 0.19 3.00 0.21 Total Expected Profit 2.26 2.38 2.44 2.37 2.18 1.97 Pr. ⇒ Profit and Exp. Pr ⇒ Expected Profit The vendors expected profit is the maximum if he buys 17 papers