Sunday, November 11, 2007

The Abused Probability (2)

This post is a continuation of a previous post.

So I was talking about how minute the chance was for me to meet Wee Loon and Violet, and how amazing it was for Richard Feynman to have seen one particular car plate. Now let me ask you a question: "Some particular thing" had happened, and the probability of that particular thing happening, if you were to calculate it beforehand, is 1 over 8.1 × 10^67, or in decimal places, 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 012.

Would you believe that "that particular thing" has just happened? Give it a second of thought.

Most of us would be inclined to think that, hey, that's such a ridiculously small probability, if that could have happened, I could as well be struck dead by an asteroid in the next minute. And if we compare it to the chance of winning top price in TOTO, we can't help but to shake our belief further in the occurrence of "that particular thing" - most of us can't even win the top price in TOTO (a chance of 1/10000), now what about that zero point zero zero zero blah blah blah probability?

The problem is, that zero point zero zero zero blah blah blah thing has just happened in front of me.

What happened was, I took out a deck of 52 playing cards, shuffled it a few times, and observed the order. It was five of heart, nine of diamond, queen of diamond, four of spade, ... , and so on. The chance of me getting that particular order, would have been 1 over 52!, which was the minute probability I mentioned in the beginning.

As you come to this point, you may start to wonder, if something with a probability of zero point zero zero blah blah blah would still happen, then what do we trust probability for? What's the point of calculating probability at all?

Some of you may have noticed that I have been calculating all the probabilities in hindsight - I calculated the chance of meeting Wee Loon and Violet after meeting them; Richard Feynman expressed his amazement at seeing a car plate after seeing a car plate; and I told you the probability of a particular order of playing cards after shuffling it and seeing the result. And if you have already suspected about it, you are in the right - this is the fundamental problem with the abuse of probability, as I have been demonstrating blatantly in the three examples.

Let's look at the example of Wee Loon and Violet. I calculated my chance of going to Safeway supermarket, their chance of going to Safeway, my chance of going at that particular hour, and their chance of going at that particular hour. If you have noticed, the probability that I ended up with is the probability of "Chang Yang meets Wee Loon and Violet at a particular time on a particular day in Safeway". Or in other words, if you simply think, "hmmmm, what is the probability of Chang Yang seeing WL and Violet at around 5pm in Safeway today?", then 1/15000 would be the correct answer, assuming that the probabilities in the calculation are correct.

However, when I did the calculation earlier, I have been doing it in hindsight. When I calculated the probability of seeing WL and Violet in Safeway, in my calculation I included my chance of going to Safeway on that particular day and in that particular hour although I was already in Safeway to begin with. That was ridiculous, because you don't go about analysing "what is the probability of myself reading Chang Yang's blog right now" when you are doing it at this very moment. So in the case of my "random encounter with WL and Violet", the chance of my going to Safeway would have been redundant. So the next time when I am in Safeway, if you go about asking me "Hey, what do you think is the chance of seeing WL and Violet here right now?", I would redo the calculation as follows: 1 / {(chance of them coming to Safeway today) * (chance of them coming at this hour)}. And that would render the chance higher than 1/15000.

For the case of Richard Feynman seeing a car plate, it's another case of "hey what's the chance of myself reading Chang Yang's blog right now" kind of thing. You don't go about asking probability of things that have happened - it doesn't mean much at all.

For example, I might as well ask, "What is the chance of Chang Yang typing a post about probability in Melbourne, at 11.30am on 11 November 2007, using Mozilla Firefox 2.0.0.9 on Windows XP, on an IBM laptop while munching on Smith Chip with barbeque flavour?". And, as silly as I could go, I would go on and be amazed, "See, such an unlikely thing has happened!"

You don't even need a calculator to see where this would lead us to.

For me to be doing the combo above, I need to be in Melbourne, I need to be using IBM laptop, I need to be having Smith Chip with me. Furthermore, I obviously need to be born to my parents, my parents need to be born to my grandparents, my grandparents need to be born to my great grandparents, and so on. And they all need to be born with the exact genetic make-up, for if they are not what they really are, then my parents wouldn't be the same people as I know them, and this Chang Yang might not have been the Chang Yang that you know. In all fairness Chang Yang could have been a Mat Rempit who has just broken his neck yesterday night from a midnight sprint.

Now we all could see that analysing the probability of things post-hoc (in hindsight) is usually meaningless. Everything that has happened had an infinitesimal probability, but that doesn't mean that it's a wonder that they happened, nor does that lend any support to skeptics who do not believe in the occurrence of the thing.

You might think that all these rambling are purely academic discussions, and that it doesn't have much to do with real life at all. You might think that for people who misuse or abuse the probability, the worst that could happen is that they underestimate their chance of meeting someone in Safeway. However, probability is abused more often and more seriously than we realize, and I shall elaborate on that in the last part of this rambling later.

[To be continued...]
[11 Nov: Continued here]