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I took two classes of Probability, I did very well and was confident on the subject. Now I meet it again in my Combinatorial Algorithm course, and guess what? I feel completely blank again! I have to review all my notes, look for examples ... even though I've already "solved" them thoroughly. I realize the amount of time I spent for these Probability classes is usually double/triple the time I spent for any other science classes. Why is it so difficult to grasp the concepts in Probability in the first place?

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Mathematical formalism? –  bgins Apr 28 '12 at 12:25
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You mean, more than number theory? –  Did Apr 28 '12 at 12:48
    
@Didier: yeah number theory is difficult too, but I still think Probability is really tricky during the exam period. –  Chan Apr 28 '12 at 13:06
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This question seems very specific to you: you seem to be asking why you find probability so difficult. (I don't believe it is universally the case that everyone finds probability more difficult than any other subject.) But before I vote to close the question as too localized, I would rather suggest you change it to something more concrete, like "How should I understand [specific topic $X$ in probability that you find confusing] so that I remember it correctly the first time?" –  Rahul Apr 28 '12 at 16:34
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Dear Chan, don't worry: probability theory is very counterintuitive and billions of dollars are voluntarily contributed yearly to lotteries, casinos and gambling houses as a conseequence! As long as it is axiomatically considered alongside measure theory, there are no particular problems. However if you want to guess results intuitively in complicated combinatorial situations, you are bound to make egregious mistakes. In my experience the only people who have a great intuition and don't make mistakes are specialists, guided by hundreds of calculations they made before (to be continued) –  Georges Elencwajg Apr 28 '12 at 18:22
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Human beings are notoriously bad at calculating probabilities.

Probability theory was not part of mathematics for 2000 years and it took more than 400 years and the genius of Cardano, Pascal, Fermat, Huygens, De Moivre, Laplace and finally Kolmogoroff to show that it could after all be integrated completely into mathematics. To give just one example, the difficulty of probability theory was spectacularly illustrated by the incredible number of mathematicians who gave wrong answers to the Monty Hall problem, Paul Erdős (one of the most amazing mathematicians of the twentieth century) being one of them.

The essential rôle of probability theory in quantum mechanics is probably also the reason why, as Bohr, Feynman and others liked to emphasize, nobody really understands quantum mechanics at the gut level (even though most of our technology nowadays derives from it). Some scientists believe that our brains developed a superb intuitive understanding of kinematics and dynamics thanks to the pressure of the environment on our hunter-gatherer ancestors (think throwing spears at sabre-toothed tigers!) but that probability theory was useless there.

To come back to your question, it might be closed because some users will feel that it is inappropriate for this site, but I am not sure that evolutionary psychology sites or similar ones will be able to answer it either. But you should try your luck (and if you do learn something there, let me know: I find the subject interesting!)

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+1 for a very insightful answer. I took 2 undergrad probability and statistics courses simultaneously with several graduate math courses that were so much more difficult-and I did vastly better in the advanced courses.And I had an outstanding teacher in Stefan Ralescu, too. It wasn't him,that's for sure. –  Mathemagician1234 Apr 28 '12 at 18:39
    
Once again, may thanks for your profound thought. –  Chan Apr 29 '12 at 5:45
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Its actually a way of thinking that is so quintessential to solve almost all text book problems in permutation and combination,

When you are doing it everyday it all seem obvious, but if you let loose of it, and then come back after some time, then it all seems distant and blurred.

Though this is somewhat true of all disciplines in mathematics , but its more evident in case of this particular subject of permutation and combination

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That's exactly what I felt! –  Chan Apr 29 '12 at 5:46
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