
The Cost of Not Understanding Probability Theory - acangiano
http://math-blog.com/2009/08/24/the-cost-of-not-understanding-probability-theory/
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tjic
_if we’ve just observed the coin appear as heads 29 times in a row, what are
the odds that the same coin will land on heads on the 30th toss?_

 _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.

~~~
crystalis
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?

~~~
tjic
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.

~~~
frossie
_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.

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yummyfajitas
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.

~~~
dvvarf
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.

~~~
yummyfajitas
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.

~~~
jerf
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.)

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RiderOfGiraffes
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.

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cousin_it
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/>

~~~
nkurz
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.

~~~
cousin_it
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.

~~~
nkurz
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.

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DannoHung
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.

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MicahWedemeyer
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.

