Perhaps...

A Chinese version of this article was posted on my personal blog some two years ago. I like it so much that I have decided to translate it into English and use it as the first post on this blog. You can tell that this blog is going to be academic (or nerdy, since I'm in graduate school), not like the random ramblings on my other blog. If you are not interested in Finance/Econ/Math/Physics/Logic/Philosophy/History, stop reading and do NOT come back again. However, don't be scared/happy, I won't post research ideas or journal papers on here. It'll be more like a scientific discussion of anything that interests me.


Probability is my favorite topic. Let me start off by giving you a typical probability problem.

Problem There are 10 people in the queue. Two of them are named A and B. Suppose all possible queues are equally likely, e.g. the queues XXXAXXXBXX and XBXXAXXXXX carry the same probability (where X denotes the other 8 people). Also suppose that each person's position is independent of everyone else's (except that they cannot be in the same position in a particular queue), and everyone is equally likely to end up in a certain position (this is called the independently and identically distributed, or iid, assumption). What is the probability of having A lining up in front of B in this queue?

Solution Combinatorics approach: Count all the favorable events, e.g. A ranked 1st, B ranked 2nd, 3rd, ..., 10th; A ranked 2nd, B ranked 3rd, 4th, ..., 10th; etc., and then divide this by the number of possible queues, 10! (This is ten factorial, i.e. 10 x 9 x 8 x ... x 1, not TEN). This works nicely, doesn't it? But what if we have 100 people instead? 100! is astronomically large.

Alternative Solution A fifth-grader might have guessed the answer correctly. The probability is simply 0.5. Note that the event "A in front of B" and the event "B in front of A" are equally likely (they are iid), and the two events make up the entire probability space (it is either A in front of B or B in front of A). Probability axioms tell us that the probability has to be 0.5.

(Side note: If there is no reason to believe that one outcome is more likely than the other, then they must be treated as equally likely outcomes. This is the Laplace's Principle of Insufficient Reason, which tells us the probability of getting heads on coin tosses is 0.5 too.)

So, what does 0.5 mean? The most appropriate answer is: If you have a large number of these queues, say 10000, law of large numbers says approximately half of them (5000) will have A in front of B.

What if I only have one queue? How do I know if I see A first or B first? The most appropriate answer is: Who knows? The sample size (one) is too small for applying the law of large numbers.

Also note that the queues themselves are independent, which is the usual and logical assumption. That is, even if we have 10000 of these queues, the arrangements of the first 9999 queues have nothing to do with the 10000th queue. You never know how a particular queue behaves, despite that you have a large sample.

In real life, how often do we have a large sample (10000 queues) and only want to be approximately correct on the macro-level (knowing that about 5000 of them have A in front of B)? Under most circumstances, we are interested to know if something will occur or not (A in front of B in this queue). Sorry, probability is not able to handle this at all. Everywhere we can find examples of highly probable events not occurring, and seemingly improbable events happening. No one knows who decides what will/will not happen.

What makes things worse is the Heisenburg Uncertainty Principle. It states that you cannot simultaneously measure the momentum and the position of an electron. Roughly speaking, you cannot be 100% certain about something by just observing/measuring other things.

So what should we do when everything in the world is uncertain and probability doesn't help much? Nothing. That's why many people resort to God/supernatural forces/fate. Maybe they are right...

To make life easier, instead of calculating probability all the time, why not just "let it be"; instead of calling something "very likely," why not just say "perhaps." Isn't the world so much simpler and nicer?