Drunkard's Math

I've picked up Mlodinow's book The Drunkard's Walk off my TBR and started in on it as I was falling asleep. True to form, when I got pissed off, I woke up. (Sex scenes, on the other hand, usually put me out like a light, right at the start. I have not run across those yet in this book.) Chapter 2 starts with the same example quoted by Malcolm Gladwell in Outliers. Basically, you're given the history of a fictional woman Linda, and asked to rank a list of possible outcomes by probability. Then they explain to you why your answer is "wrong" according to probability. It bothered me when I read it in Gladwell and it bothered me again at the source. But I'm cooling off because I've figured out why the answer and explanation both piss me off.

The three main results at hand are that Linda is:

1. active in the women's movement (AWM)
   
2. a bank teller (BT) AND active in the women's movement
   
3. a bank teller

This is the order most people list as the probable outcome based on the fact that Linda is a feminist and shows no bank teller tendencies. It's more likely that she's be AWM and stumble into banking, being the reasoning. But according to probability, two outcomes are always less likely than one outcome so BT AND AWM is less likely than either BT or AWM in isolation. Whether it's at all likely that Linda will become a bank teller, it is always less likely that she will be both a bank teller AND active in the women's movement.

The part that pissed me off was that the author's thesis is that people ignore the rules of randomness and probability to their detriment, but his answer and explanation ignored the rules of logic to his and our detriment. His explanation is something like "oh, she could have a good reason to give up on the activist stuff" making A less likely, A+B is always less likely than A or B in isolation. The issue with the "she could have had a good reason" explanation is that it's both dismissive of the student/reader and dismissive of probability. Given the inputs and the stated "correct answer", it's not helpful in explaining anything and is not likely besides. Given a test question with similar answers but no personal history to start off with, people generally order the results A, A+B, A+B+C. So what gives here?

This Caltech prof is baffled that some people cling to the notion that the order is 1,2,3 rather than 1,3,2 even when the probability is explained. I'd be one of those people and here's why. (Might mean my answer is still wrong, but here goes.*) I can tell you now it's not that "she could have a good reason to give up on the activist stuff".

If young Linda had an equal chance at either AWM or BT, we get
50% AWM,
25% AWM AND BT
50% BT
Even if we know AWM > BT and to get 50%, 40%, AWM+BT comes out lower at 20% as they explained always happens. Clearly misordered, no?

No.
First, we're given Linda's "history" and a reason to favor AWM over BT. The history is very strongly weighted to suggest AWM is very likely and BT is a shot in the dark. Given some random schmo we know nothing about, we can guess his or her probability of taking any given job with any given activity as equal for the sake of the problem. For Linda, the issue with the "correct" answer is that in A and B are not isolated in this particular problem but have a relationship to Linda's history, which we're told to consider relevant. We cannot dissociate the probabilities of these two things. (It's not clear from the discussion whether or not the test designers ever argue the probability of AWM over BT or grade down for making that assumption but I suspect not.) Her probability of AWM >> BT. So intuitively, we predict that she'd be more likely to stick with the AWM than give it up and rate AWM+BT higher than BT.

For instance, given her background, I'd rate a 95% chance of AWM and a generous 10% chance of BT. The mathematical probability AWM AND BT is then is 9.5%, less than the 10% chance of being a bank teller. So the order as explained would be

95% AWM
9.5% AWM AND BT
10% BT

Still clearly wrong, no?

No.
Second, the answers do not occur in a vacuum, so given the history input for the question and the outcome list above, the implied options are really INTERPRETED as:

1. AWM AND NOT BT
   
2. AWM AND BT
   
3. BT AND NOT AWM

because logically, if the last option was "BT and may or may not be AWM", it makes the third answer non-separable from the second answer, so the only way that BT alone could be a logical and distinct option in this list is that it means she is both a Bank Teller and NOT active in the women's movement.

Using the probabilities above (AWM 95%, BT 10%) the outcome for this more logical list is different and matches the intuition.
85.5% AWM NOT BT
9.5% AWM AND BT
0.5% BT NOT AWM

The "NOT BT" in the first statement has some effect but it's negligible to the sequencing of the rest of the list because BT << AWM means anything with +BT will be much less (<<) likely than anything with +AWM. For the same reason in reverse, "NOT AWM" has a huge effect on sequencing.

While it's entirely probable that Linda does not stay active in the women's movement because "oh, she could have a good reason to give up on the activist stuff", it remains consistently much less likely for her to become a bank teller and the relative probabilities hold. In fact, NOT AWM is more probable than BT NOT AWM at 5% over 0.5%. For completeness, and because the primary objection to the "correct" answer was that the probability of the Linda of record giving up AWM was lower than the "correct" answer implied, let's add NOT AWM, NOT BT. That leaves
85.5% AWM NOT BT
9.5% AWM AND BT
4.5% NOT AWM NOT BT
0.5% BT NOT AWM
Which is what people figured out anyway. It wasn't that the couldn't do probability, it was that the "correct" answer didn't account for the NOT factor and it should have.

So MY thesis is that the people who listed the original order got both the probability AND the logic of this problem right, but the Caltech professor got the logic wrong and thus solved the problem incorrectly because his probabilities were based on incomplete descriptions. And not because "he could have a good reason to give up on the math stuff". And given that Mlodinow chides people only a couple of pages earlier for not considering "A NOT B", it's fair to chide him for not considering it here.

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* My answer could not-unique if there's some randomly selected A and B with A > B such that the sequence changes. My answer could be incorrect if there is good reason to exclude "NOT AWM" from BT, but in the problem as given, there is not good reason to do so.

I am leaving it as an exercise to the reader to figure out at which independent percentages of A > B that the order goes from
A NOT B, A AND B, B NOT A
to
A NOT B, B NOT A, A AND B.

And thinking about it a little more, I think the question and answer are badly presented. It does show that absent a more complete understanding of the question being asked, one can get the "answer" wrong. Consider using my earlier stated probabilities, here are ALL the options in order:

95% AWM - [BT or NOT BT]
85.5% AWM NOT BT
10% BT - [AWM or NOT AWM]
9.5% AWM AND BT
4.5% NOT AWM NOT BT
0.5% BT NOT AWM

So it is the case that BT > AWM AND BT and always will be. But the question was set up to imply BT NOT AWM, not BT. I read it as "if you ran into Linda at a reunion, how surprised would you be to find she is X", each answer is unique. But each answer is not unique. If one were working on this at, say, work and only had a few options (and not 600), one could list all the possibilities like I did above and then say "are we interested in BT or BT NOT AWM? Because it matters. This does raise an interesting facet of sorting out possibilities and coming to a more complete understanding of the situation by adding in options that aren't listed in the hope of sorting through the ones that are.

But I maintain that to answer the question, one orders BT NOT AWM after BT because it is calculated from the BT NOT AWM inference. If the goal is to point out when and when not to add the "Not" into probability calculations, or the risks of doing so, the given explanation still fails.

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But to give the man his props, he did explain something really important socially in Chapter 1. Long term studies show that positive reinforcement leads to better behavior and outcomes than negative reinforcement. This is seemingly contradicted by short term observations that if you praise someone for doing a great job, their next effort is usually not as good, but if you yell at them for doing a bad job, their next effort is usually better. Here, he gets both the logic and the probability right, and gives everyone a way to explain two seemingly contradictory experiences.

The concept is performing to the norm. Your work on any given day tends to be of consistent quality, but can vary for better or worse. Your work on any given day is LIKELY to be near the mean of your usual work. Let's call this GOOD. So no matter what you did today, your work tomorrow is most likely going to be GOOD. On rare occasions, you do GREAT or BAD work that is out of the norm enough to invite comment.

If you do BAD work today and get yelled at, tomorrow's work will likely be GOOD thru sheer probability. (Unless you get so upset that you tie one on and can't work as well with a hangover.) But when you have that merely GOOD day, GOOD is still better than BAD. The conclusion reached is that the yelling worked because your work improves from BAD to GOOD.

You see where this is going? Yep. If you do GREAT work today, you're still likely to only do GOOD work tomorrow. Since you got heaps of praise for doing GREAT then only did GOOD afterward, the observation is that praise doesn't work because your work went from GREAT to GOOD which is observed as a step backward.

Over the long haul, continued praise for GREAT work, raises the level of GOOD work more than continued berating for BAD work does but those differences in GOOD are in the noise of day to day observation. So for all you folks out there cheering on your kids for getting an A and coaching them through the Ds, and cheering the dog for pooping in the right place, you're on the right track.