Question

Jose Morales manages a large outdoor fruit stand in one of the less affluent neighbourhoods of San Jose, California. To replenish his supply, Jose buys boxes of the fruit early each morning from a grower south of San Jose. About 90 percent of the boxes of fruit turn out to be of satisfactory quality, but the other 10 percent are unsatisfactory. A satisfactory box contains 80 percent excellent fruit and will earn $200 profit for Jose. An unsatisfactory box contains 30 percent excellent fruit and will produce a loss of $1,000. Before Jose decides to accept a box, he is given the opportunity to sample one piece of fruit to test whether it is excellent. Based on that sample, he then has the option of rejecting the box without paying for it. Jose wonders (1) whether he should continue buying from this grower, (2) if so, whether it is worthwhile sampling just one piece of fruit from a box, and (3) if so, whether he should be accepting or rejecting the box based on the outcome of this sampling.

Use the likelihoods/conditional and prior probabilities to calculate all the posteriors/revised probabilities. Show your work/calculations. (6 points) N. B. : Round the probabilities to two decimal points.

Draw and solve the decision tree for this problem to determine the optimal decision strategy that Jose should follow. (Show your work/calculations). Verbally communicate the decision strategy. (25 points)

Answer 1

Should Jose continue buying from this grower?

Decision alternatives: continue buying from the grower, find a new supplier

Possible outcomes: satisfactory boxes of fruit, unsatisfactory boxes of fruit

Probabilities: 0.9 for satisfactory boxes, 0.1 for unsatisfactory boxes

Expected value of continuing to buy from the grower:

EV = (0.9 x $200) + (0.1 x (-$1,000))

EV = $170

Since the expected value is positive, it is worthwhile for Jose to continue buying from the grower.

Is it worthwhile sampling just one piece of fruit from a box?

Decision alternatives: sample one piece of fruit, do not sample

Possible outcomes: good sample, bad sample

Probabilities: 0.8 for good sample, 0.2 for bad sample

Expected value of sampling one piece of fruit:

EV = (0.8 x EV(satisfactory box)) + (0.2 x EV(unsatisfactory box))

EV = (0.8 x $200) + (0.2 x (-$1,000))

EV = $40

The expected value of sampling one piece of fruit is lower than the expected value of not sampling at all. Therefore, it is not worthwhile for Jose to sample just one piece of fruit.

If Jose samples one piece of fruit and it is excellent, should he accept or reject the box?

Decision alternatives: accept the box, reject the box

Possible outcomes: satisfactory box, unsatisfactory box

Probabilities: 0.8 for satisfactory box, 0.2 for unsatisfactory box (assuming the sample is excellent)

Expected value of accepting the box:

EV = (0.8 x $200) + (0.2 x (-$1,000))

EV = $40

Expected value of rejecting the box:

EV = 0

The expected value of accepting the box is higher than the expected value of rejecting the box. Therefore, if Jose samples one piece of fruit and it is excellent, he should accept the box.

Overall, Jose should continue buying from the grower, not sample one piece of fruit, and if he does sample one piece of fruit and it is excellent, he should accept the box.

Step-by-step explanation: