There are four grocery stores in your neighborhood and you are the manager of on

Question

There are four grocery stores in your neighborhood and you are the manager of one of them. A customer has returned a relatively expensive and damaged item for a refund without a sales slip. You know that the item is carried by all four stores and you account for 65% of the sales of the item in the neighborhood. You are more careful in screening these items as you buy them, hence usually only have 0.25% defectives in your inventory. Your estimate of % defectives at your competitors is 0.3%

What is the chance that the customer bought the item from you?

If you do not give a refund, your eventual loss of good will is $70 if the person did buy the item from you. There is no loss if it was not bought from you. The future earnings from this customer are increased by $10 if he bought it from you and $40 if he did not. The cost of the refund to you is $50. On a mathematical basis, should you make the refund?

Explanation / Answer

Probability that the customer bought the item from our shop is computed as:

= Probability that the customer bought a defective one from our shop / Probability that a defective item is bought from any shop

= (0.65*0.0025)/ (0.65*0.0025 + 0.35*0.003)

= 0.6075

Therefore 0.6075 is the required probability that the customer bought the item from us.

Expected value given that the refund is made = -50 + 0.6075*10 + 40*(1-0.6075) = -28.225

Expected value given that the refund is not done = - 0.6075*70 = -42.525

Therefore the refund should be done.

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