You work for a firm that sells a product. The probability that any random person

Question

You work for a firm that sells a product. The probability that any random person wants to buy your product is 1%. A marketing firm claims that it will be 90% accurate in identifying the people who will buy your product and, also, 90% accurate in identifying the people who will not buy your product. They want to sell you the names of the people they have identified as potential buyers for $1 per name. Should you purchase $1000 names? Your product sells for $10.

Of the 1000 people who would buy the product, how many will the firm indicate as "likely to buy" (LTB)?

In other words, what is P(LTB | Buy)?

  Of the 99,000 who will not buy the product, how many will the firm indicate as "likely to buy" (LTB)?  

In other words, what is P(LTB | No Buy)?

Explanation / Answer

Solution:

This is similar to an epidemiology question, where "buy" is the "disease," the firm's assessment of likelihood the "test," with a sensitivity of .90 and a specificity of .90.

So we can make a table summarizing the results with a prevalence of .01. Let B represent "buy," B' "not buy," L "likely buyer," and L' "unlikely buyer.

There are 1000 buyers. With a sensitivity of .90, then 900 are identified, so if a subject buys the item, there's a .90 probability that the subject scored as a likely buyer.

There are 99,000 non-buyers, so with a specificity of .90, then 89,100 (90% of 99,000) of the non-buyers will be identified as unlikely by the test.

But here is where the problem comes in: 10% of the non-buyers will be identified as likely buyers, since p(L|B') is .10. This is 9900 non-buyers identified as likely buyers. Only 900 of the likely buyers were identified as such, so p(B|L), the probability that a "likely" buyer will actually buy, is only 900/9900, or 1/10.

So the positive predictive power of the "test" is very small. This is due to the .01 prevalency rate. Such a small proportion of the subjects would buy, even very high sensitivity and specificity rates do a poor job of predicting whether a subject would buy:

P(L|B) = P(L'|B') = .90 looks good, but in reality you are not "given" B or B'. You are given L and L' and want to ESTIMATE P(B) and B' based on L and L'.

P(B|L) = .10; P(B'|L) = .90. Not a good bargain. You'd be buying 90% erroneous predictions.

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