- Which is more likely -- that a mother with blue eyes has a daughter with blue eyes or that a daughter with blue eyes has a mother with blue eyes?
- Are there more words beginning with the letter 'k' than with 'k' as the third letter?
- Smoking increases the risk of lung cancer by a factor of ten and of fatal heart disease by a factor of two: do more smokers die of lung cancer than of heart disease?
- Do more people die of strokes than of accidents?
- Which is more dangerous, cycling or riding the Big Wheel?
- Consider two maternity hospitals, one averaging forty-five births a day, the other fifteen: in which hospital is it more likely that on any given day 60 per cent of births will be boys?
Thursday, June 14, 2007
Irrationality
In the introduction to his book, Stuart Sutherland offers some sample questions to demonstrate to the reader that rational thinking is sporadic at best. Some are subjective ("Are you a better than average driver?") -- here are the more objective questions:
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Maybe I'm really dumb, but I think these are trick questions.
1 and 3 don't give you enough information to answer the question.
2 is a stupid question that's easily answered by running grep on a dictionary file - how does it have anything to do with reasoning?.
4 and 5 are comparing a very common activity with a rare "risky" activity, tempting you to pick the "risky" one, which we all know is a fallacy.
What's missing from question 6 is that the 60% figure is way off the true ratio (which is around 52/48). So, highly anomalous figures are more likely with a small N, but 15 and 45 are both pretty small Ns.
"Not enough information to answer" is a perfectly rational response. I'm too much of a sucker for puzzles so I gave answers for all six. Here they are (with some spoilers):
1. It's more likely that a daughter with blue eyes has a mother with blue eyes. I can't remember why I thought this.
2. I said there are more words that begin with 'k'. Sutherland uses this to illustrate availability bias because it's easier to think of k... words than --k... words. He says there are more words with 'k' (also 'r') in the third position than in the first.
3. Knowing that heart disease kills more people than lung cancer, I figured it was a trick question and answered that more smokers die of heart disease.
4. Accidents kill more than strokes.
5. Cycling is obviously more dangerous.
6. Like beowulf, I think a small N leads to a greater likelihood of the 60% result, so I picked the hospital with about 15 births.
For #6 I don't know why you wouldn't just calculate it. Assume 60% means exactly 60%, and assume PB = the probability of a boy = 50%. Then the total number of unique sex distributions for N babies is 2^N, and the total number of ways to get 60% boys is (N C 0.6*N) ["N choose 0.6*N]. So the probability P is (N C 0.6*N) / 2^N.
Err, I guess I shouldn't assume any knowledge of probability. "a choose b" is the number of ways of choosing b objects from a set of a objects, and is equal to a! / (b! * (a - b)!).
So if N = 15, we have P = 15! / (9! * 6! * (2^15)) ~= 15.3%.
If N = 45 then P = 45! / (27! * 18! * (2^45)) ~= 4.9%.
Wouldn't your calculation be effected by the fact that the probability of having a live birth be a boy is not 50%?
http://en.wikipedia.org/wiki/Sex_ratio
That's why I said "assume it's 50%".
The calculation is more complicated if you use 52.5%, but it won't change the sign of the difference in probability.
Although, dang, it's not intuitively obvious how to alter my calculation to use a different PB, so I'm now kind of doubting my method.
Without being told which country the hospitals are in, I think assuming a 50% ratio is justifiable. It's not that far from the world average of about 51.2% (105 boys out of 205 babies -- where did 52.5 come from?).
I think a more significant problem with calculating #6 is that we're told the hospitals only average 15 and 45 births per day. If the actual number of births is not a multiple of 5, there can't be an even 60% division. That's why I thought the 60% figure was not intended to be exact either, and the only factor we should consider is that larger samples will tend to be closer to the average, so the smaller hospital is more likely to have an anomaly. I would think that principle would be known to more than just the Rational Elite, but Sutherland may have evidence to the contrary.
51.2% (105 boys out of 205 babies -- where did 52.5 come from?).
From dumbth. I divided 105 by 200. More evidence that mathematicians (if I can still claim that title in these latter days of numeracy) make the worst arithmeticians.
Of course, I stipulated that I was making assumptions to get a rough figure.
It occurred to me last night that there may be enough information to answer 1.
Blue eyes are a recessive trait so there is a 100% chance a blue-eyed daughter has a blue-eyed mother.
The eye-color that is manifested in a child is only partially dependent upon the color of the mother's eyes. If the father is blue-eyed then there is a 100% chance the mother will have a blue-eyed child. If the father is something other than blue-eyed then there is a 25% chance the mother will have a blue-eyed child.
However, only 49ish% of children are daughters so the odds of the blue-eyed mother having a blue-eyed daughter are always less than the odds of the blue-eyed daughter having a blue-eyed mother.
> Blue eyes are a recessive trait so there is a
> 100% chance a blue-eyed daughter has a blue-eyed
> mother.
If blue eyes are a recessive trait, doesn't that mean a daughter could be blue-eyed even though _neither_ of her parents are? They would both have to be _carriers_, but that's not the same thing.
Oy. You are correct. Both parents could be heterozygous (Bb) and have a blue-eyed daughter.
Halfway through the book, the author has yet to shed light on the other questions, but I already have my doubts about his authority. The following two passages made me wonder about his rationality (and, alas, he wrote "I beg readers not to inform me of any errors they detect in this book" in the preface).
"A study conducted by Fortune magazine found that over 50 per cent of American businessmen thought that following tradition was more important than making profits." (p. 62)
I couldn't find any evidence of this study with its dubious conclusion, and the book gives no citation.
"In 1974 thirty-six investment advisers were asked to pick their five favourite stocks. There was a large degree of consensus, but when the performance of the most favoured ten stocks was examined over the two-year period 1972-73, it was found that the recommended stocks had dropped by 27 per cent more than the average fall during the declining market then prevailing." (p. 64, emphasis added)
Odd, isn't it?
Worse, his assertion about words starting with K appears to be simply false. A reliable online wordlist gives the following counts for words from three to ten letters:
k-- 87
--k 68
k--- 482
--k- 210
k---- 1089
--k-- 461
k----- 1442
--k--- 501
k------ 1287
--k---- 506
k------- 1067
--k----- 475
k-------- 718
--k------ 296
k--------- 451
--k------- 175
Overall, the total is 6623 words beginning with K and 2692 with K as the third letter.
I'm not sure I understand the objection about the investment advisors. It seems like Sutherland is saying "Why did these dudes pick these 10 stocks, when they just got finished sucking?" Of course, that's not really a reason to question the picks - if they are strong stocks historically, it's probably a good time to buy just after they dip. Or was that your point?
Yes, it would seem much more relevant to look at the performance of the stocks after they were picked.
I don't think I agree. What you suggest would demonstrate their skill at prognostication; looking at the pre-pick performance would seem to better demonstrate their rationality.
Making money by correctly predicting which stocks will rise in value, regardless of their past performance, seems an eminently rational behavior for investors. Given the well-known tendency of stocks to show erratic patterns, simplistically choosing stocks because they have been rising seems, most charitably, to violate the venerable buy-low-sell-high principle. We might do well to look up how the best performers of the past two years did in the previous two years. Maybe uncanuck can advise us?
Making money by correctly predicting which stocks will rise in value, regardless of their past performance, seems an eminently rational behavior for investors.
Sure, but they can't know ex ante which stocks will win. Being wrong != being irrational. All else being equal, I'd say it's easier to judge rationality by looking at the period before the picks than after.
Of course, buying low is not irrational, but it's an understandable, if not forgivable in a book on rationality, mistake for Sutherland to make to assume that picking stocks that just got done sucking is an error.
In other words: having good (bad) reasons for one's picks bears more on rationality than the actual result.
If I am presented with the Monte Hall Problem, switch, and lose, was I irrational?
I think the rationality of a stock pick is tangential to its performance which is probably similar to what ray's getting at.
Was it a good performing pick? is a different question from Did you have a good reason for the pick (i.e., was the pick rational)?
The performance of the stock before or after may be offered as support but is not necessarily a reliable determiner of reasonable picks.
But then the pragmatist in me has to ask, who cares if the pick was reasonable, if the picker performs poorly over the long term?
It's better to be right than reasonable in the real world.
...for those (hypothetical) situations where a "right" vs "reasonable" dichotomy exists.
Sutherland describes an experiment that purports to demonstrate people's tendency to look for evidence that supports their beliefs rather than evidence that might contradict them. This is irrational because no amount of evidence can ever prove a theory, but a single counterexample can disprove it.
In the experiment, subjects were told that the series 2, 4, 6 conforms to a certain rule. They had to figure out what the rule is by stating additional series of numbers and learning whether they conformed to the rule.
I did a highly scientific version of this experiment on six subjects via instant message. To make a game of it, I had people try to guess the rule in the minimum number of turns, one turn consisting of either:
1) stating another series of three numbers, and the tester would tell them whether it conformed to the rule, or
2) guessing the rule.
Here are the results:
Subject R
1. 18, 20, 22
2. consecutive even integers?
3. 4, 6, 8
4. -6, -4, -2
5. -2, -4, -6
6. -2, 0, -2
7. 1, 3, 5
8. n, n + 2, n + 4?
9. 4, 2, 6
10. 1, 50, 8000
11. k, m > k, n > m?
self-assessment: "11 is shameful"
Subject D
1. 4, 8, 12
2. 3, 6, 9
3. x, 2x, 3x?
4. 2, 3, 4
5. 1000, 2000, 3000
6. 3, 2, 1
7. 1, 3, 2
8. x, y > x, z > y?
self-assessment: "wow, i suck"
Subject M
1. 8, 10, 12
2. ascending even numbers?
3. 14, 16, 18
4. 3, 5, 7
5. 2, 8, 10
6. 1, 2, 3
7. 5, 4, 3
8. numbers of ascending value?
self-assessment: "spock himself would be proud"
Subject G
1. professor plum, the lounge, lead pipe!
1. 11, 22, 33
2. so do i win?
2. ok -- the second number is twice the first, the third is thrice the first. oui?
3. 6, 4, 2
4. 1, 2, 3
5. 1, 1, 1
6. 2, 3, 4
7. numbers listed in ascending order?
self-assessment: "i thought i was logalicious."
Subject E
1. 3, 5, 7
2. 3, 5, 10
3. 9, 4, 5
4. 66, 67, 68
5. ascending numbers?
self-assessment: "probably about average or slightly better"
Subject S
1. 7, 9, 11
2. 5, 2, 1
3. 7, 8, 1007
4. any number greater than the previous number?
self-assessment: "i've heard of this test before"
I have it on good authority that subject E's first guess was "100, 102, 104," meaning he got it in 6, not 5.
You're right, the transcript shows that E did take six turns. Oh, and I just made up the "highly scientific" bit.
From page 87:
"Here is another ingenious but straightforward problem that many people find difficult. You are told to imagine that you have a candle, a cigarette lighter and a box of tacks and are asked how, using just these three objects, you would attach the candle to a wall. Pause to think of the answer. If you get it in less than three or four minutes you are unusual."
The funny thing here is, within the first minute I thought of two ways to attach the candle to the wall, but neither way requires all three objects. I could do it without the lighter, or without the tacks, but haven't yet thought of a way to use both.
Okay, I've come up with a way to use all three. You could use the lighter to soften the wax, and then embed the heads of the tacks in the candle, so that the candle would have spikes on it and could be stuck into the wall. But it would be simpler just to melt the wax with the lighter and then stick the candle directly to the wall, or to tack the box to the wall and then set the candle in it (maybe with a row of tacks around the base and some guide-tacks at top and sides to keep it standing).
Melt the wax until you have a good length of wick, then tack the wick to the wall.
The rather disappointing answer from the author: "The solution is to empty out the tacks, burn some wax off the candle with the lighter and use the molten was to attach the empty box to the wall; finally the candle is placed upright in the box being held steady by wax burnt off its base." MacGyver would never consider such a rickety solution, and Eric's direct approach seemed the most elegant to me.
On the question about blue eye inheritance, the author says that "three subjects in four thought it was more probable for the mother to have a daughter with blue eyes," because "people display more confidence in reasoning from cause to effect than from effect to cause." The original research he cited can be seen by searching for "blue eyes" in Amazon's copy of Judgment under Uncertainty: Heuristics and Biases, or see Example 1. This source suggests that both cases are equally likely, and the correct answer (which Sutherland omits) was in fact the most popular answer.
There were no surprises in the author's treatment of smokers' risks, strokes/accidents, and cycling vs. amusement park attractions. On the maternity hospital question, he followed Car Talk practice and left out essential details when posing the problem. Subjects were told that roughly as many girls were born as boys over the course of a year, and asked which hospital would have more days on which 60% of the births were boys. "Most of them thought there would be no difference."
That box-and-candle solution isn't as good as any of the ones we came up with. I suppose Sutherland's point was that most people don't pay attention to the existence of the box, and just think about the tacks? (They don't "think outside the box"). If so, he needs a better example.
My wife came up with the same solutions I did for using the lighter only, and for using both the lighter and the box of tacks (but not for using the box of tacks alone). The three-number-sequence test didn't work as well on her:
1) Counting by twos?
2) Ascending even numbers?
3) I give up.
They say women prefer concrete problem-solving.
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