The Abused Probability (1)

Probability, also known as chance or odds, is simply the ratio of one thing happening divided by all the other possible scenarios for a particular incident. For example, if there are two boxes, one of which contains a surprise, the probability of me opening a box which contains a surprise would be one out of two. Simple and straightforward.

As someone with an undying interest in mathematics, I am disheartened by the constant abuse of the simple concept of probability in various areas. It is thrown around up, down, left and right by both people who know maths and people who thought they know maths.

Let's begin with a simple example.

One day, at about 6pm, I walked down to the local Safeway supermarket and met Wee Loon and Violet. I was surprised to see them in Safeway, as I had never bumped into them before. As I saw him, I started thinking, "Wow, that's a coincidence! What in the world is the probability of me bumping into Wee Loon and Violet in Safeway today?"

In order to bump into Wee Loon and Violet in Safeway, obviously, first I have to go to the Safeway supermarket. Since I don't go to Safeway often, I reckon that would be a probability of, say, one in ten of me going to Safeway in a particular day. But, Wee Loon and Violet don't go to Safeway that often too since they are busy with studies. So, say, for that particular day, the chance of them going to Safeway is one in fifteen.

So the chance of me going to Safeway on the same day with them is one in 150.

But hey, that's only the chance of us going to Safeway for today!! I could have gone in at 4pm and they gone in at 5pm, and we could still miss each other! So I have to take the time into consideration. So what is the probability of myself going to Safeway at 6pm? Probably one out of twenty. And what is Wee Loon and Violet's chance of being in Safeway at 6pm too? Probably one out of five.

So if you do the calculation, the chance of me bumping into Wee Loon and Violet in Safeway at 6pm that day was about... 1/ (150 x 20 x 5) = 1 in 15,000!! Just imagine the coincidence! 1 in 15,000, that's even less than the chance of getting the top number in TOTO!

Something must be wrong here. But what? I have already been under-estimating the probabilities above, so the probability could have been lower.

At this point, we might look at another example by the maverick physicist, Richard Feynman:

“You know, the most amazing thing happened to me tonight. I was coming here, on the way to the lecture, and I came in through the parking lot. And you won’t believe what happened. I saw a car with the license plate ARW 357. Can you imagine? Of all the millions of license plates in the state, what was the chance that I would see that particular one tonight? Amazing!”

I hope that at this point you have realized the flaw of both mine and Richard Feynman's arguments. Mathematically both the examples above were almost flawless, I followed through the various probability of the multiple conditions, and arrived at the final answer with standard mathematical methods. Mr. Feynman's mathematics was impeccable too. There indeed were millions of license plates in the state, and he indeed only saw that particular fated number.

As you might have noticed, the problem here is not with the maths - the problem here is with the framework in which I performed the calculation, which I will expound on later.

[To be continued...] (sorry sophisticatedsoul)
[11 Nov: Continued here]