Statistics question: I dare you Bayes or binomial?

Question

3. Assume that you are a dealer in PC and have hired a new sales agent for a 2-week trial period. At the end of the two weeks, you must decide either to release him or place him permanently on the payroll. Based upon past experience, you estimate that given your prospective sales agents’ experience, personality and other traits, there is a 20 percent probability he will be a superior sales agent, a 70 percent probability he will be an average sales agent, and 10 percent probability that he will be an inferior sales agent. A superior sales agent is one who sells a PC to 40 percent of his customers, an average sales agent sells to 30 percent of his customers, and an inferior sales agent sells to 20 percent of his customers. After the 2-week period, you note that he has demonstrated PC to a total of 20 customers and has sold PC to 8 of them. Based upon all the information you have, should you retain the man permanently in your employment if your policy is to employ only average or su

Answer 1

P(superior) = 0.2
P(average) = 0.7
P(inferior) = 0.1
P(sells 8 out of 20 | superior) = 20C8 (0.4)^8 (0.6)^12 = 0.1797 (4 d.p.)
P(sells 8 out of 20 | average) = 20C8 (0.3)^8 (0.7)^12 = 0.1144 (4 d.p.)
P(sells 8 out of 20 | inferior) = 20C8 (0.2)^8 (0.8)^12 = 0.0222 (4 d.p.)

P(sells 8 out of 20) = 0.1797 (0.2) + 0.1144 (0.7) + 0.0222 (0.1) = 0.1182 (4 d.p.)

so P(superior | sells 8 out of 20) = 0.1797 (0.2) / 0.1182 = 0.304 (3 d.p.)
P(average | sells 8 out of 20) = 0.677 (3 d.p.)
P(inferior | sells 8 out of 20) = 0.019 (3 d.p.)

So the probability that he is average or superior is 0.304 + 0.677 = 0.981 = 98.1%; you should retain him.

Answer 2

Since his pre-employment profile indicated that his likelihood of being average or better was 90% and since he did perform at a 40% rate for two weeks I would definitely grant him permanent employment.

Answer 3

yes the probability is that he will continue to sell at a superior level even if his sales fell an order of magnitude he would still be a good risk.