In Steven Pinker's "How the Mind Works" (Norton, 1997) he gives many 
examples of what the human mind does easily and what it does poorly. A 
theme running through the book is that since homo sapiens evolved from 
tree living primates that moved onto the savannahs, our minds are highly 
evolved for that environment but do less well at tasks that arise from 
situations very foreign to such life. In his chapter 5 he discusses the 
mind's perceptions of probability, or more accurately likelihood, since 
early man did not think of it quantitatively. He writes:

 "The founders of probability theory, like the founders of logic, assumed 
 they were just formulating common sense......But then why do people often 
 seem to be 'probability blind', in the words of Massimo 
 Pilasttelli-Palmarini? Many mathematicians and scientists have bemoaned 
 the innumeacy of ordinary people when they reason about risk. The 
 psychologists Amos Tversky and Daniel Kahneman have amassed ingenious 
 demonstrations of how people's intuitive grasp of chance appears to flout 
 the elementarty canons of probability theory. Here are some famous 
 examples:"

[Part of the explanation for this is that humans evolved in an environment 
containing few examples of *independent random events* that were of 
concern to them. Thus, if the weather has been bad for a series of days, 
they learned to *expect* that the likelihood of good weather increased 
during that period. This is totally unlike the case of flipping a coin in 
which the events are statistically independent, but people *expect* that 
if they get lots of heads in a row, the likelihood of a tail is 
increased.]

The examples that particularly caught my attention is this:

 "This problem was given to sixty students and staff members at Harvard
 Medical School: 'If a test to detect a disease whose prevalence is
 1/1000 has a false positive rate of 5%, what is the chance that a person 
 found to have a positive result actually has the disease, assuming you
 know nothing about the person's symptoms or signs?' The most popular 
 answer was 0.95. The average answer was 0.56. The correct answer is 0.02, 
 and only eighteen percent of the experts guessed it."

Can you confirm that alleged "correct answer"?

7 responses total.

Ok, I'm not a math whiz, but out of an average group of 1000 people, 1
person will have the disease. But 50 people in the group will test
positive for it and not really have it. One divided by 50 is 0.02, but
that one person with the disease can't be both infected by the disease
and have been falsely tested positive. The two conditions are mutually
exclusive. So it should be 1/51, which comes to 0.0196.

Thanks for joining in.....your answer is correct, but only by making an
additional assumption. The problem is unsolvable as stated. One has to know
what the probability is that a person with the disease will test positive.

If we assume that is 1.0, then the exact probability is 1/50.95, very 
close to your answer. But that might not be. If the probability of getting 
a positive if the person does have the disease is also 5% - that is, the 
positive rate is 5% whether a person has the disease or not, the 
probability that a person testing positive has the disease is 0.001, the 
same as the prevalence.

(The problem is one involving what is known as Bayesian estimation.)

This one cames from _A Mathematician Reads the Newspaper_ by Allen Paulos, 
and concerns on how in real situations random choices make make more sense 
that deterministic choices. Here's the situation:

"A pitcher and a batter are facing each other. The pitcher can throw 
either a fast ball or a curve ball. If the batter is prepared for a fast 
ball, he averages 0.500 against such pitches but, thus prepared, he only 
averages 0.100 against a curve ball. If he's prepared for a curve ball, 
however, the batter averages 0.400 against them, but he only averages 
0.200 against fast balls in this case.

"Based on these probabilities, the pitcher must decide which pitch to 
throw and the batter must anticipate this decision and prepare 
accordingly."

Since any deterministic strategy could be analyzed and countered, the best 
strategy for both is to choose completely randomly (and independently) to 
throw a fast or curve ball, and to prepare randomly (and independently) 
for each. The question is, with what probability should the pitcher choose 
to throw a fast ball to minimize the probability of the batter getting a 
hit while at the same time with what probability should the batter choose 
to prepare for a fast ball to maximize his probability of getting a hit?

The author gives the answer but completely omits the *mathematics*, which 
I thought would be the main content of the book. His point, though, is 
that there are many situations where the best strategy is to flip a coin - 
or choose a random number - rather than create some deterministic strategy 
- in labor conflicts, market battles, sports (as here), in war, etc.

Since he didn't give the math for calculating the answer, I spent a good 
part of the evening doing it. It against demonstrated to me that good 
perceptions of probability are not inherent in our psyches.

I was hoping that some baseball aficionados, who like to think about 
sports probabilities, would join in. This problem has turned out to be 
much more interesting than the author made it in the cited book.

The answer given by that author is that the pitcher should throw pitches 
that are randomly fast balls and curve balls with a probability of 0.5, 
and the batter should prep for fast balls 1/3 of the time and for curve 
balls 2/3 of the time, but chosen randomly also. With the hit 
probabilities given in the question, the probability of a hit is just 
0.300. However I have looked at the probability of a hit in the full 
sample space for the throwing and hitting strategies of the pitcher and 
batter, and it turns out that:

1. At the optimum point for the pitcher given above, 0.5, the probability 
of a hit does not depend upon the batter's choice of his probability of 
preparing for a fast ball - is remains the same, 0.300, for the batter 
preparing for a fast ball none of the time to all of the time.

2. Likewise, at the optimum point for the batter, 1.3, the probability of 
a hit does not depend upon the pitcher's choice of his probability of 
throwing a fast ball - it remains the same, 0.300, for the pitcher 
throwing a fast ball none of the time to all of the time.

3. HOWEVER - if the pitcher increased his probability of throwing a fast 
ball from 0.5 to, say, 0.6, and the batter detects this, then the optimum 
strategy for the batter is to prepare for ALL fast balls, in which case 
his probability of getting a hit increases to 0.340.

4. LIKEWISE - if the batter increased his probability of preparing for a 
fast ball from 1/3 to, say, 0.400, and the pitcher detects this, then the 
optimum strategy for the pitcher is to throw ALL curves, in which case the 
probability of the batter getting a hit decreases to 0.280.

Are any of these results obvious to baseball fans, given the probabilities 
originally stated?

(I realize that the conditions of the problem statement are unrealistic, 
as it assumes the same batter and pitcher are continually "up", and that 
those probabilities remain the same, and that it is even possible for a 
batter to "prepare" for a fast vs curve ball -- but then, this was a 
contrived example for the author's thesis, that a random strategy in some 
cases may be better than a deterministic strategy.)

There is another consequence of the pitcher, say, increasing the fraction 
of fast balls a little, causing the batter to prepare only for fast balls. 
The pitcher will notice that and immediately switch to all curve balls. 
But then the batter will notice that and immediately switch to preparing 
for all curve balls, and this cycling will continue. The "optimum" 
situation appears to be unstable. Which makes this a lousy example of what 
the author Paulos was trying to illustrate - that there is an "optimum" 
probabalistic strategy.

So the example he gave is lousy, but he gives no more realistic examples 
from the other fields he cites: "business (labor conflicts and market 
battles), sports (virtually all competitive contests), and the military 
(war games) that can be modeled this way".

Maybe he meant the randomized decision process was best over a large number
of games? If the players have equal intelligence, they should be able to
counter any decision strategy either could come up with, so over a large
number of games their wins would seem even. Then again I wonder if the pitcher
has an advantage anyway because the batter couldn't immediately react to the
pitcher's strategies as he came up with them; it seems like the batter would
always be a few pitches behind as he tried to figure out the pitcher's
strategy.

That would be true if the pitcher cannot distinguish the batter preparing for
a fast or curve ball. The batter, of course, can more easily distinguish a
fast or curve throw. 

These choices were, by the terms of the problem, all randomized, so it is true
that it takes a larger number of throws, if not games, for the probabilities
to become evident. 

Response not possible - You must register and login before posting.

- Backtalk version 1.3.30 - Copyright 1996-2006, Jan Wolter and Steve Weiss