The Cost of Not Understanding Probability Theory(math-blog.com)
math-blog.com
The Cost of Not Understanding Probability Theory
http://math-blog.com/2009/08/24/the-cost-of-not-understanding-probability-theory/
6 comments
You're nitpicking unnecessarily. When asked a 'do the trains collide?' math question, do you point out all the safety precautions in place that would prevent the collision?
You're missing my point.
The original post was arguing "people are stupid and credulous".
I am arguing that, no PEOPLE ARE NOT STUPID - they are finely evolved to operate in the real world. If you present people with a REAL WORLD PROBLEM and they come to EXACTLY THE RIGHT ANSWER (as opposed to the "desired" theoretical answer) then the problem is with your ability to quiz people, not with people failing to be smart.
The original post was arguing "people are stupid and credulous".
I am arguing that, no PEOPLE ARE NOT STUPID - they are finely evolved to operate in the real world. If you present people with a REAL WORLD PROBLEM and they come to EXACTLY THE RIGHT ANSWER (as opposed to the "desired" theoretical answer) then the problem is with your ability to quiz people, not with people failing to be smart.
You're missing my point
I think you are missing your own point. Faced with 99 head tosses, you think the 100th should be heads because it proves the system is gamed (eg. the coin is biased).
The article addresses people who believe that faced with 99 head tosses, they think the 100th should be tails (because it is "due").
In other words, the people in the article behave as if they believe that the system in independent (not gamed) but reach the wrong answer.
I think you are missing your own point. Faced with 99 head tosses, you think the 100th should be heads because it proves the system is gamed (eg. the coin is biased).
The article addresses people who believe that faced with 99 head tosses, they think the 100th should be tails (because it is "due").
In other words, the people in the article behave as if they believe that the system in independent (not gamed) but reach the wrong answer.
But your argument supports the opposite of the classic gambler's fallacy. The traditional mistake is to expect the 30th flip to be less likely to be heads, because 30 in a row is rare. So peoples' intuition is even less appropriate to the real world than probability theory is.
Desired theoretical answer: 1/2.
Your answer: bet on heads
"EXACTLY THE RIGHT ANSWER": Less than a one in a billion chance of heads.
Your answer: bet on heads
"EXACTLY THE RIGHT ANSWER": Less than a one in a billion chance of heads.
_IF_ the draws are independent, it's not bullshit and you just lost a lot of money. In the example around which the article is written the draws are very much independent.
I saw somewhere a similar article with pictures of "randomness". The point was it's very counter-intuitive: random events tend to look very un-evenly distributed. Here lots of zeroes, here lots of ones. Given the big number of "draws" in the Lottery, it's not at all unusual to see a number appear or not appear for a while. So you don't even have reason to suspect foul play.
I saw somewhere a similar article with pictures of "randomness". The point was it's very counter-intuitive: random events tend to look very un-evenly distributed. Here lots of zeroes, here lots of ones. Given the big number of "draws" in the Lottery, it's not at all unusual to see a number appear or not appear for a while. So you don't even have reason to suspect foul play.
> _IF_ the draws are independent
That's a big IF.
Let's do an experiment with people: we ask them to stand in a road, then we tell them that in a moment a hologram of a fire truck will come at them. They are requested to stand in the road. 30 seconds later a massive firetruck appears to race at them, lights going, siren blaring. Sunlight reflects off of the chrome. The guy in the driver seat is wearing a helmet and a black jacket, and scowling. The license plate is "113 GHR".
The experimental subject steps out of the road.
"Dummy!" we yell... we TOLD YOU that it was a hologram.
The candidate isn't a dummy. He just trusted his experience and his own sense a lot more than he trusts one word from you.
That's a big IF.
Let's do an experiment with people: we ask them to stand in a road, then we tell them that in a moment a hologram of a fire truck will come at them. They are requested to stand in the road. 30 seconds later a massive firetruck appears to race at them, lights going, siren blaring. Sunlight reflects off of the chrome. The guy in the driver seat is wearing a helmet and a black jacket, and scowling. The license plate is "113 GHR".
The experimental subject steps out of the road.
"Dummy!" we yell... we TOLD YOU that it was a hologram.
The candidate isn't a dummy. He just trusted his experience and his own sense a lot more than he trusts one word from you.
And your point is? This is Occam's Razor. It's much more likely that it's a real truck. Also the stake vs the gain is too big.
I'm talking about a state organized lotterie. Given the huge risks involved in cheating, the fact that the profit margin is assured even without cheating and the fact that there is no one person or entity who really cares about bigger profits (state owned and managed), I'd say the same Occam's Razor would say the draws are very likely independent.
I'm talking about a state organized lotterie. Given the huge risks involved in cheating, the fact that the profit margin is assured even without cheating and the fact that there is no one person or entity who really cares about bigger profits (state owned and managed), I'd say the same Occam's Razor would say the draws are very likely independent.
Such a great illustration for the article.
Using coin flips gives a terrible example. The odds are actually pretty high that the coin will be heads next.
Suppose there are 1 billion coins in circulation, 100 of which are double sided [1] (due to magicians accidentally spending them, etc). Then:
P(double sided) = 10^{-7}.
P(30 heads in a row) = 10^{-7} + (1-10^{-7}) x 9.31 x 10^{-10} ~= 10^{-7}
P(double sided | 30 heads in a row) = P(30 heads in a row | double sided) x P(double sided) / P(30 heads in a row) = 1 x 10^{-7}/ 10^{-7} = 1
(Via Bayes rule.) So the coin is probably double sided and the gambler is right. The mathematician who lacks common sense is wrong.
Suppose there are 1 billion coins in circulation, 100 of which are double sided [1] (due to magicians accidentally spending them, etc). Then:
P(double sided) = 10^{-7}.
P(30 heads in a row) = 10^{-7} + (1-10^{-7}) x 9.31 x 10^{-10} ~= 10^{-7}
P(double sided | 30 heads in a row) = P(30 heads in a row | double sided) x P(double sided) / P(30 heads in a row) = 1 x 10^{-7}/ 10^{-7} = 1
(Via Bayes rule.) So the coin is probably double sided and the gambler is right. The mathematician who lacks common sense is wrong.
Amusing comment, however a fair coin is not double sided by definition. :)
In any case, I changed the example from 30 to 10 tosses, so that we avoid "missing the forest for the trees".
In any case, I changed the example from 30 to 10 tosses, so that we avoid "missing the forest for the trees".
But what does a 'fair coin' have to do with bets in the real world? Assume I'm standing on a street corner, flipping a coin . You observe 30 flips, and note that 30 times in a row it comes up heads. I then turn and offer to bet you 2:1 odds that the next flip will also be heads. Would you really take my bet?
No, you would walk away, despite your superior knowledge of probability. But let's say that it was only 10 flips in a row, as in your modified example. Would you stay and bet? If you would, perhaps it is you is is missing the forest. :)
More precisely, perhaps the problem is not their knowledge of probability, but the difficulty of modeling the real world with standard probability theory. The Bayesian approach above at least allows for the possibility that the coin is not fair. How does your model allow for this?
No, you would walk away, despite your superior knowledge of probability. But let's say that it was only 10 flips in a row, as in your modified example. Would you stay and bet? If you would, perhaps it is you is is missing the forest. :)
More precisely, perhaps the problem is not their knowledge of probability, but the difficulty of modeling the real world with standard probability theory. The Bayesian approach above at least allows for the possibility that the coin is not fair. How does your model allow for this?
When does the author talk about double sided coins? The example is based on the premise that the coin is fair and each toss is independent of the others.
I'm disputing the premise. The author has a prior, namely "fair coin". I'm suggesting that it's likely that the prior is wrong.
So, what's the alternative? Give up the concept of "fair coin" when trying to teach statistics? This seems unlikely to be a useful pedagogical move! How is anyone supposed to understand Bayes Theorum before they even understand probability?
Mathematicians have to be able to specify priors. Insisting they are unrealistic rather misses the point that the entire reason the phrase "fair coin" appears in the first place is an explicit acknowledgment of the unreality of the situation.
If you want to more rapidly move to a Bayesian system when teaching, I'd understand completely, but we still have to start with "fair coins" before we can get around to considering how to prove they aren't fair. (I will be first in line to say many school teachers, being generally poorly trained in mathematics, completely fail to understand this point, and correspondingly ask the "gotcha" question about "what's after 29 heads?" without fully understanding the issue themselves.)
Mathematicians have to be able to specify priors. Insisting they are unrealistic rather misses the point that the entire reason the phrase "fair coin" appears in the first place is an explicit acknowledgment of the unreality of the situation.
If you want to more rapidly move to a Bayesian system when teaching, I'd understand completely, but we still have to start with "fair coins" before we can get around to considering how to prove they aren't fair. (I will be first in line to say many school teachers, being generally poorly trained in mathematics, completely fail to understand this point, and correspondingly ask the "gotcha" question about "what's after 29 heads?" without fully understanding the issue themselves.)
The point is that common sense forms the basis of our prior knowledge. If your priors are poor, then your probability estimates are going to be inaccurate, even if you know probability theory perfectly.
The chances that there are two bombs independently on a plane are utterly, utterly miniscule, and that's why when I travel I always carry a bomb with me. I can be certain there won't be another one, and since I won't let mine explode, I'm safe.
And so are you.
And so are you.
Yep, the gambler's fallacy is quite well known. The "heuristics and biases" field of psychology thinks it's caused by the representativeness heuristic.
http://en.wikipedia.org/wiki/Gamblers_fallacy
http://en.wikipedia.org/wiki/Representativeness_heuristic
Minor quibble: if a coin fell heads 29 times in a row, my credence for it falling heads a 30th time is higher than 0.5 because I now have reason to believe the coin is biased. How much higher depends on how many biased coins there are, as yummyfajitas said.
Here's another epic example of people failing to use basic probability theory: http://lesswrong.com/lw/13i/shut_up_and_guess/
http://en.wikipedia.org/wiki/Gamblers_fallacy
http://en.wikipedia.org/wiki/Representativeness_heuristic
Minor quibble: if a coin fell heads 29 times in a row, my credence for it falling heads a 30th time is higher than 0.5 because I now have reason to believe the coin is biased. How much higher depends on how many biased coins there are, as yummyfajitas said.
Here's another epic example of people failing to use basic probability theory: http://lesswrong.com/lw/13i/shut_up_and_guess/
I'll quibble with your link as well. Did Yvain actually test to see whether his guessing increased his score over the semester? Or did he just 'prove' it?
Like the presumption of a fair coin in this page, his presumption is that the questions are fair. If they were trick questions, such that the seemingly improbable answer is correct more often than not, guessing based on partial knowledge may not be a benefit.
I think he's right, but I'd trust his results more than his abstract reasoning.
Like the presumption of a fair coin in this page, his presumption is that the questions are fair. If they were trick questions, such that the seemingly improbable answer is correct more often than not, guessing based on partial knowledge may not be a benefit.
I think he's right, but I'd trust his results more than his abstract reasoning.
Yes, guessing based on partial knowledge might make you worse off. But as Yvain's footnote 1 says, replacing 30 "I don't know" answers with 30 purely random coin-flip guesses has a 98% chance of improving your score. (I'd like to go on record saying emphatically that this statement doesn't need to be "tested". Recalculated, at most.) So if you have a coin in your pocket, it's pretty hard to rationalize answering "I don't know" to many questions instead of flipping the coin.
I appreciate your response and voted it up. But I disagree that there is no benefit to actually testing his hypothesis. The mathematics don't need to be 'tested', but perhaps the priors do? For example, how certain are you of the assumption that the questions that we need to flip the coin for are evenly distributed between true and false?
It seems plausible that a test might have a bias whereby the harder questions are predominately one answer or the other. How large would this effect have to be to overwhelm the 2% probability? Would you trust your grade on the presumption that this effect can be ignored? What about your life?
For the record, I agree with Yvain's advice. I did just fine on the SAT's, partly because I followed the mathematically correct advice that one should guess if one can eliminate at least one of the 4 choices. But I wouldn't proclaim that this strategy needs no testing just because the mathematics are correct.
It seems plausible that a test might have a bias whereby the harder questions are predominately one answer or the other. How large would this effect have to be to overwhelm the 2% probability? Would you trust your grade on the presumption that this effect can be ignored? What about your life?
For the record, I agree with Yvain's advice. I did just fine on the SAT's, partly because I followed the mathematically correct advice that one should guess if one can eliminate at least one of the 4 choices. But I wouldn't proclaim that this strategy needs no testing just because the mathematics are correct.
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On some tests you get a zero score for not answering and a negative score for answering wrong. Balanced in such a way that uniform randomness gives you the same expected score as not answering.
The difference between trying to determine events statistically and performing an empirical experiment is very important for understanding why people believe in this fallacy.
If you are physically observing a coin land heads up ten times in a row, you might expect that the mechanism for flipping the coin is so regular that it is repeating the same non-relativistic physical interaction every time.
The thought experiment fundamentally relies on people reconstructing the physical experiment in there head. If you instead described a physical experiment where the randomness was more prevalent (and the fairness of the coin was demonstrated explicitly), I think fewer people would believe the fallacy.
If you are physically observing a coin land heads up ten times in a row, you might expect that the mechanism for flipping the coin is so regular that it is repeating the same non-relativistic physical interaction every time.
The thought experiment fundamentally relies on people reconstructing the physical experiment in there head. If you instead described a physical experiment where the randomness was more prevalent (and the fairness of the coin was demonstrated explicitly), I think fewer people would believe the fallacy.
Understanding something and applying that knowledge are two very different things. Plenty of perfectly sensible people have strange superstitions that defy all common sense.
I won't argue that teaching better probability skills would not be beneficial to society. Still, it's not going to stop people from making stupid bets. Some people just have a crazy need to gamble.
I won't argue that teaching better probability skills would not be beneficial to society. Still, it's not going to stop people from making stupid bets. Some people just have a crazy need to gamble.
Many people would argue that the chance of this happening is less than one in a billion, as we just calculated. However, that answer is blatantly wrong.
Bullshit.
If a coin lands heads 29 times in a row, Bayesian analysis tells us that the chance of the coin not being fair, the experimenter lying to us, aliens influencing the outcome, etc. are all a lot higher than we might have expected a priori.
If I saw a coin land head 29 times in a row and was offered a bet on the next flip, I'd put some real money on it turning up heads.