# Thinking Probabilistically

From: Stevan Harnad (harnad@cogsci.soton.ac.uk)
Date: Sun Jan 28 1996 - 22:27:05 GMT

Here's a summary of the last seminar. Take a look at the Koehler papers,
and the Tversky & Kahnemann if you're especially interested!

We all have intuitions about probability. Some of them are right and
some of them are wrong. The "image" of probability is a big urn with
white and black marbles, say, half and half. You reach in and take out,
say, ten marbles without looking, count how many are black and how many
are white, then put them all back in. (That's called "sampling with
replacement.")

Now the question: If, as with the 6-button sandwich machine we talked
about a few weeks ago, your daily lunch depended on your saying exactly
how many white marbles there were out of the ten each time, how do you
make sure you get lunch the most often, and how often would that be?

Answer: Always guess 5/10, and that way you will eat the most often,
but I made a mistake in class: It will not be, on average, every second
day. It will be less often than that. I'd need a table of the binary
distribution to find out exactly how often, but here's why it's less
than half the time: If you were taking out only one marble every time,
and predicting black or white, then it would be just like tossing a
coin, with only two possibilities, and you would indeed eat eat, on
average, half the time. But with the urn, on any one sample of 10 at a
time, the possibilities are of course more than just two: it could be
1/10, 2/10, ... 9/10, 10/10. The most frequent case will still be 5/10,
but it won't happen half the time, but less often. So 5/10 is still
your best bet. The point of the example is that, even where an outcome
is random, there are ways to bet that maximise your chances.

Then I think I gave the example of the obstetrician who makes money-back
guarantees to predict the gender of your baby before it's born (and
before the era of in utero imaging): His best strategy is to predict all one
gender -- and in fact it's better to predict boys only, because slightly
more than 50% of births is boys (but they don't survive quite as often
as girls). Even with the money-back guarantee, he'd make money half the
time -- and even a bit more.

That little bit more is what gives the gambling casino the edge over
clients -- except card counters (people who have a system for keeping
track of the cards that have already been dealt: that's like sampling
from the urn WITHOUT replacement, so you can predict better than chance
what's left, based on what's already been removed). And that's why they
get barred from casinos when they are discovered.

I asked also when under what conditions it was rational to buy a lottery
ticket DAILY if the prize was 100 pounds and there were 100 tickets each
time: Answer: if they cost a bit less than a pound each. Then in the
long run you'd always come out ahead. Otherwise, you'd just break even
or lose. But of course the Camelot lottery is not like that!

I also mentioned that it was illegal to double your bets in casinos, for
obvious reasons: Suppose you're betting 100 pounds on a coin toss. You
lose, but you're allowed to bet 200 on the next coin toss. You lose
again (so far you lost 300) so you double and bet 400 on the next one.
You win! (Eventually you would have one, because of the odds.) So you
pocket the 100 you made and start again. Patiently, you'd eventually
keep adding the amount of you initial bet to your pocket, if you had
enough money to keep doubling (and the Casino allowed you) in between.
But they don't allow you...

Why does doubling work? Because in the long run the averages always
prevail, if the coin is fair.

So I then asked: Supposing there is a FAIR coin and before your eyes you
see it tossed 19 times and each time it comes out heads: You bet now,
what do you bet on? More than half the seminar said heads, though
Nik changed his mind when I reminded him the coin was fair. You would of
course lose money if you were ready to bet any more than the usual 50/50
that the next one would be heads. "It doesn't matter" is the right
answer, because there are no "winning streaks" that you can grab a hold
of. Some people would be ready to bet more on tails, because they think
the coin is dues for a tails after all those heads, since it has to come
out even in the end. But the coin has no memory and doesn't care about
how things come out in the end! There are lots of coins, tossed lots of
times; sometimes they'll come out 19 heads in a row by chance; sometimes
even more! It's just like occasionally pulling out more or less than
exactly 5 white marbles from a half/half urn: It happens, but you can't
bank on it; you can only bank on the average.

Then we got to the case of the disease (From Tversky, A. &; Kahneman,
D. (1982) Evidential impact of base rates. In Judgment under
uncertainty: Heuristics and biases, (pp. 153-160), eds.
D. Kahneman, P. Slovic, &; A. Tversky. Cambridge University Press)

You have a medical symptom; you go to the doctor. He says it's serious,
it's a bacteria, it's fatal if untreated, but if treated, it's curable.
However, the bacteria comes in two strains, a common strain (85% of the
time it's that one), called A and a rarer strain (15% of the time it's that
one), called B. You need to decide which one you want to be treated for,
because the treatment for one does not work for the other and vice
versa, and you only have enough time to treat for one. Obviously you'd
want to choose A, though with a certain amount of nervousness.

But wait, says the doctor, there IS another test I can do, to see
whether you've got the A or B kind. So you take the test, and the result
says you've got B, the rare kind. The test itself, is 80% reliable:
8 times out of 10, it gets the strain right.

Question: Which one do you ask to be treated for? Most people say B,
because that was MY test, it was about ME, whereas those others
statistics, about how often it tends to be A or B in general are not
about me.

But the fact is that those base rates about how often it tends to be A
and B ARE about you, and if you did the calculations, you would find
that even if the test said you had B, your chances are still better to
be treated for A (so probably it was a bad idea to take the test at all,
since it could only make you more nervous).

The correct calculation is based on Bayes' rule, for calculating
conditional probabilities (the probability that you will have A given
that the test says you have B). This calculation is complicated, and
most people simply ignore the base rate for irrational reasons, so it is
said but see:

http:/cogsci.soton.ac.uk/~bbs/Archive/bbs.koehler.html
or
ftp://cogsci.soton.ac.uk/pub/harnad/Psycoloquy/1993.volume.4/psyc.93.4.49.base-rate.1.koehler

In the case of the gambler's fallacy (thinking the coin has some sort of
memory that makes things average out), the error is in the intuition
that you can defy the averages based on what you know of the history of
the coin. In the case of the Kahneman & Tversky baserate fallacy the
error is ignoring the population baserates. Now the well-known Monte
Hall Paradox, which seems to go exactly against the CORRECT intuition
when you see the coin has no memory, and it makes no difference if you
bet heads or tails after a series of 19 heads:

You are in a quiz show. There are three curtains. One conceals a big
prize. You can choose any of the three curtains. (To stick to our
principle that none of this makes sense on a one-time basis, you should
imagine yourself doing this every day, and the prize is lunch.) If the
lunch is hidden randomly, you would eat, on average 1/3 of the time.

Now the announcer gives you another chance (as with the test for the
disease): He knows where the prize is, so after you have made your
choice, he opens one of the other two curtains, always one where the
prize ISN'T. Now, the question is: Given a second chance, do you (1) stick
with your choice, (2) switch to the other unopened curtain, or (3) it
doesn't matter?

Most people say, reasoning as they do with the coin, that it doesn't
matter, but it does! For if you stick to your original choice, it is
clear that in the long run you are "married" to a 1/3 chance. But
opening one curtain has now changed the odds to 1/2 -- IF you choose
randomly between the remaining two. That's already better then 1/3.
But you can do even better if you switch, for the one you chose is
"married" to 1/3, the one that was revealed is out of the running, so
the third curtain actually has the remaining 2/3 chance of being right!

Sounds like magic? No, you have to remember that, doing this over and
over, you are getting INFORMATION by being told where the lunch ISN'T.
You must make use of that information or be condemned to the ignorance
that the 1/3 guess represents.

Then came Donald McKay's "Brain-Reading" Machine about free will and
determinism. I had forgotten to say that the famous mathematician
Laplace, thinking he was being completely logical about cause and
effect and determinism, and the predictive laws of physics, said that
if he knew the position and momentum (speed/mass/direction) of every
particle in the universe right now, then he could predict everything
that would ever happen. He was wrong, because the interactions are too
complex to be calculated exactly, except for two-way interactions, so
his predictions could just be statistical, like a weather report.

But never mind that. Suppose someone had a complete "scan" of everything
going on in your brain right now, and using that, could correctly
predict everything you would do no matter what happened. (Note: Because
of Laplace, he cannot predict what happens in the rest of the world, but
he CAN predict exactly what you would do for anything that might
happen.)

MacKay, a believer, said you would still have free will if there were
such a machine, because, if told its predictions, you could always
decide to do otherwise (Oh yeah? You predict I'll say yes? Well then I
say "no"). Trouble is, that if the premise is true, that the machine can
predict it all, it can also predict what you will do if told its
prediction. That's a prediction too, and it's always one step ahead of
you. If you COULD do anything that machine didn't predict, that would
simply show it couldn't predict, whereas the ASSUMPTION here is that it
can predict it all, with 100% accuracy.

No problem here: MacKay is simply wrong in thinking there would be any
room left for your free will. But then came the last puzzle, where there
is no right or wrong answer:

Again, a machine that can read your brain state so completely that it
can correctly predict anything you will do under any conditions. It has
taken its reading, and based on its results has done the following:

It has placed 100 pounds underneath a transparent globe. Beside it,
under another globe that you cannot see through, it has placed either
1000 pounds or nothing, based on the following rule, based in turn on
what it had (correctly) predicted you would do: If it correctly
predicted that you would be "greedy" and take what was under BOTH bowls,
it put nothing under the opaque bowl. If it predicted you would be
"temperate," and would voluntarily forego the 100 pounds that you could
see, and take only what was under the opaque bowl, then it put 1000
pounds under the opaque bowl.

But that is all history. What's done has been done. You now face the
bowls and can do what you like: What do you do, and why?

Recall that the premise is that the machine has correctly read you mind
and can predict everything you do EXACTLY (not statistically, which
would make all this much easier, since you could do it over and over):

Most people say they would go for both bowls because it would be
superstitious to give up the 100 pounds given that whatever had happened
had already happened. This is the choice that rejects backwards
causation in time as absurd.

Some people say they forego the 100 pounds because their faith in the
power of truth -- on the assumption that the premise that the machine
never errs is true -- is "stronger" than their rejection of backwards
causation. Either way, it's a bit like choosing who would win in a
supercontest between an immovable force and an irresistible power!
There is no correct answer.

But what if the game were iterated, i.e., if the machines predictive
powers were only statistical, and you could play the game every day?
What would you do then, and why?

This is related to the iterated prisoner's dilemma, which I will discuss
next time...

Chrs, Stevan

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