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If ever there was a case where this become clear is with the Monty Hall problem. Even the great Paul Erdos got fooled by this problem. My question which may be difficult to answer is what is it about probability that we can be so confident of an answer we get uwing an intuitive argument and yet be so wrong. Benford law on first digits and the waiting time paradox are other famous examples like this.

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well, the Monty Hall problem seems to be a situation where people think, a priori, that whenever there are two possible outcomes, they are equally likely. Another example is the concept that an outcome is "due" in a sequence of independent trials. In any case I'm not sure these fallacies you refer to are properties of probability (re: "what is it about probability...") but rather involuntary simplifications people make. – Macro May 5 '12 at 4:26
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We can trust our intuition, but we need to practice to have the right intuition. – Stéphane Laurent May 5 '12 at 8:07
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Some of our false intuitions are due to heuristics and related concepts. See a lot of work by Kahnemann and Tversky. Some of these heuristics make sense, evolutionarily. – Peter Flom May 5 '12 at 11:11
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As a general rule, and it's to be illustrated in the Monty Hall problem, it seems like most probability mistakes are made due to mistakes in conditional properties, or, often ignoring conditioning altogether. – Gschneider May 5 '12 at 14:27
@Gschneider I think you hit the nail on the head. – Michael Chernick May 5 '12 at 15:52
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There are two main approaches to understanding this question. The first (and I believe most successful) is the literature on cognitive biases (see this LessWrong link).

Much has been written on this topic and it would be too presumptuous to summarize it here. In general, this just means that the cognitive machinery humans are endowed with through the evolutionary process employs lots of heuristics and shortcuts to make survival decisions more efficiently. These survival decisions mostly applied to ancestral environments which we rarely face anymore, and so the frequency with which we face scenarios where our heuristics fail might be expected to increase.

Humans, for example, are great at generating beliefs. If positing a new belief costs very little, but failing to employ a belief that would have lead to survival has high cost (even if the belief is in general incorrect), then one would expect to see lots of rationalization and low evidence barriers to believing propositions (which is what we do see with humans). You also get behaviors such as probability matching for similar reasons.

One could go on at length describing all of the fascinating ways that we deviate from opimal decision making. Check Kahneman's recent book Thinking, Fast and Slow and Dan Ariely's book Predictably Irrational for popular, readable accounts with lots of examples. I recommend reading some of the sequences at LessWrong for more principled discussion of cognitive bias, and lots of interesting academic literature reviews regarding steps one can take to avoid these biases in certain circumstances.

The other approach to this problem is (I think) far more tenuous. This is the notion that probability is itself not the correct normative theory for dealing with uncertainty. I don't have time to annotate some sources for this now, but I will update my answer later with some discussion of this view.

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Someone downvoted me on this question. I would like to know why? – Michael Chernick May 7 '12 at 2:19
It wasn't me. My guess is that they think it is not in line with the FAQ guidelines. Probably they think it doesn't have a specific enough answer, or that it's not concretely related to statistics. I think it's a fine question, and I don't know why someone would downvote. – EMS May 7 '12 at 2:24
Thanks for the explanation. I wasn't trying to point fingers. Two people voted me up for the question. I think this site should concentrate more on asking good questions and giving good answers. I see such a fixation with the rules. – Michael Chernick May 7 '12 at 2:33
I agree and I think it applies generally across several stack exchange sites. The moderators sometimes wield too much power to close a question. Even when many people up-vote, it often only takes the negative attention of 3-4 moderators to get a question closed. I think questions should be innocent until proven guilty, so to speak, and a long, drawn-out discussion on the meta sites should be required, with tons of votes to close, before the question can be closed. User votes naturally do away with bad questions, so there's no real danger of too many "junk" questions. – EMS May 7 '12 at 3:34

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