The question is should John now conduct the test on another bulb from Box 2? Wha

Question

The question is should John now conduct the test on another bulb from Box 2?

What is now John's optimal decision policy and certainty equivalent?

John is an exchange student at SUN University. His room light has just burned out, and so he goes to the only store in the neighborhood to buy a replacement bulb. John requires a bulb for only six more months. In the store, John finds two boxes of light bulbs: In Box 1 are bulbs that are guaranteed to last at least 6 months, selling for $30 each. If John buys a bulb from Box 1 he can return to store for a free replacement if it burns out within six months. In Box 2 are bulbs with no warrantee, selling for $10 each. The store owner explains that all of the bulbs in Box 2 are of the same type, and it is equally likely that they are either of Type A or of Type B. Type A bulb has a 0.5 probability that its lifetime is less than 6 months while Type B bulb has a 0.75 probability that its lifetime is less than 6 months. The store intends to dispose of Box 2 soon. Hence, if John buys a bulb from Box 2 and it burns out within 6 months, he must return and buy a bulb from Box 1

Explanation / Answer

Here the case is as follows:

Case 1: John buys bulb from Box 1 and pays $30 and gets an assurance of having a bulb for 6 months. In this case his total investment wil be 30$

Case 2: If he buys a bulb from Box 2 which has A and B type of bulbs. Now the proportion of a and B type of Bulbs in box 2 is not given so we take a as the proportion of bulbs of A category and b as the proportion of bulbs of B category. Category A has 0.5 probability of failure while B has 0.75 probability of failure so the probability of selecting a faulty bulb is (0.5a+0.75b)/(a+b) and so the investment here will be

10 +[(0.5a+0.75b)/(a+b)] * 30 -------- eq(1)

The first part of the above equation (italic) explains the cost associated with buying bulb of type 2 while the second part of the above equation (in Bold) gives the cost associated with the faulty bulb multiplied with the probability of bulb being faulty.

Since we don't know the proportion of a and b we can' answer regarding the choice of Box 1 or 2 but as it looks from the question that here they are assuming the probability of a and b as 0.5 each and by substituting we get the ans of a=b=0.5 we get

10 +[(0.5a+0.75b)/(a+b)] * 30 = 10+18.75=28.75$ which is less than 30$ of cost from the case 1 and so we can conclude that John should now buy the Bulb from Box 2 and yes he should look at the Box 2 and so he need to test the bulb. So the answer is John should now conduct the test on another bulb from Box 2 since he is now planning to buy a bulb from Box 2.

I hope the explainantion has helped you understanding the concept. Kindly upvote if it has really helped you. Good Luck!!

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