Bob Lochel asks: “I am wondering if anyone can conjecture why “3” was featured in both answers, as I can’t wrap my head around it.  Perhaps it was a counting of the number of cells in the row she needed?”

• mpershan

I’m not exactly sure what the deal with the 3 is, but it just has to have something to do with the numbers 2 (“not bothered”) and 25 (“total males under 40”).

Maybe the second answer just mimicked the denominator of the first? (There is visibly less struggle on the second problem than in the first part.)

• There are 3 columns. The numerator represents a quantity from the first column. Once the author settled on 3 as the denominator this remained consistent throughout. Clearly a lack of understanding of probability.

• Jamie Roberts

Was there a previous example in class or in the packet where the denominator was 3?

• This was actually a college course, so we did not do many examples. But the one class example dealt with a similar-looking table, and there was a textbook reading. Neither place features the number 3 prominently.

• The numerators are also off. In the first problem, they selected the “bothered” column instead of the “not bothered” one. In the second problem, they didn’t add in the females over 40, demonstrating that they look to the table for answer instead of using common sense to extract meaningful information.

Incidentally, you get the correct answers using Bayes’ Theorem, which was my first impulse. But finding each term for it requires knowing how to read the table.

• Huh… yeah, baffling. I teach probability, but this is a new one on me! The very fact that the question SAYS ‘Calculate the probability’ means the denominator must, by definition, be the total. So the fact that it’s a *3*… not even the total of all totals… really, as said, all I can think is they saw three columns and so divided by total number of columns.

I SUPPOSE, if we reach, the correct answer IS 0.08 (or 8%). And 7.6 (760%) IS close to 8. Maybe they had some intuition, and just grabbed a denominator that gave them something like 8? Then, as was said, once they were satisfied with the poor total for (a), they carried it over to (b), so that’s… something.

What’s peculiar is that it looks a bit like they first wrote ’43’ as the denominator. (Hard to tell in the scan.) The wrong branch, implying confusion of events A and B, but at least a total. So a better question might be why they thought a probability of over 100% made more sense than what they had before.

Aside: This should probably be filed under “Conditional Probability”.

• I would be interested in hearing what the student’s take was on this one, as the idea of using ‘3’ makes absolutely no sense in this question.

• Jim Doherty

I’ve used this exact problem myself. I think I got it from Chris Olsen when he was my summer Stats Institute teacher. A few BIG problems here. Most obvious is the problem with a student being comfortable writing fractions that are greater than 1 as answers to any probability question. The first answer uses the number of males under 40 who are bothered rather than those not bothered as the problem indicates. The student repeats the same mistake in part B so there is some real consistency here. Same denominator, same mistake in the numerator. I know that it is somewhat standard to concentrate on complements when trying to find certain probabilities so that might explain the numerator choice. The denominator? I think that the theory of using the number of columns as the denominator might be a little too simple but it may also be at the heart of the problem here.