Friday, November 09, 2007

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]


owl said...

huh, was reading blogs randomly and chanced upon my name...hehe:D

sophisticatedsoul said...

Not another incomplete post..but well, apology accepted. :P

I was looking up on probability/game theory recently (inspired by Deal or No Deal) and now you blog about abused probability. What a coincidence?! But no, I won't start calculating the chance of this.

WP said...

Oh, nice example! That's why I hate probabilities. There are so many people in the world and the probability of bumping into one of them is so small. But, we still bump into so many people each day! ;P

bluez_aspic said...

Pr(the sequence of events having happened)= 1

Still, having a Chor Dai Dee hand comprising of 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A, 2 is pretty woot - something which I can lay claim to =)

Oh, and here's a nice anecdote where the rule unambiguously preceded the coincidence:

Shoblast was horrified to discover how many mutual friends we shared in Melbourne (and this was before we had contact of any sort), so out of exasperation he randomly picked someone from his msn list and asked whether she knew me.

"Yeah, he's my neighbour."

Now, THAT was amzing.

Weijian said...

Haha interesting. If Feynman had predicted the number before he actually saw it, that would be amazing. Likewise, only if you have foreseen that you would bump into Wee Loon and Violet before the evening...Lol you seriously can win the TOTO already.

Hey continue with your post!! =) I look forward to reading the explanation. Is it something like, Feynman would surely see a car, so the chance is 1, not 1 over some amazingly big number? Haha nice post but perhaps the first sentence should be "Probability, also known as chance or odds, is simply the ratio of one thing happening divided by all possible scenarios for a particular incident" by removing 'the other'. Your sentence makes sense only when n is sufficiently large =P

Eric Fu said...

Despite being a future mathematician, I do not fancy the field of probability, at all. I cannot imagine myself being a probabilist. That's partially why I decided to abandon the actuarial career path.

Nonetheless, a well-written post!

youngyew said...

OWL: Hah so what's the probability of that happening eh? :P

Sophisticatedsoul: Eerm, talking about "Deal or No Deal", I recently read about an interesting maths question which is quite relevant to the game:

Two players play a game of "number guessing". The first player will write ten numbers on separate pieces of paper, and those number are between 1 to 10^100. The number are then covered. The second player will now do the guessing. He uncovers the number one by one, and he will win if he comes to a particular number, decides that it is the largest number among all, and gets it right.

Now, what is the optimum strategy for the second player, and if he uses the same strategy all the time, what is the long-term probability of him winning the game?

WP: Haha yeah, when we bump into someone we know we always think "hey that's fate / karma / whatever mythical element that decides such an encounter". However, if you think about it, the chance of us meeting "someone we know" on an unspecified street isn't that slim, given that we all live in pretty densely populated area and most people hang out at the same place for shopping, entertainment etc.

bluez: With your "luck" of having such a great hand of big two, I can say wit 100% confidence that you must have played quite a lot of big two throughout your life! :D And also, you know a lot of people.

But that's pretty coincidental, the examples that you quoted.

Weijian: Is that you Chuah Wei Jian? I am surprised to see you here! :D

Yeah you knew what I was going to write about, it's basically about calculating probability of things that have happened.

And yeah, you are right, I should have said "one thing divided by all things" instead of "one thing divided by all other things". Sharp eyes eh. :P

Eric Fu: Hah I can remember how you always disliked probability, like how I never fancied calculus. :P

Hey in fact I can still remember one of the Form 5 probability questions where we supposedly "cheated" right before the exams. It's about a T-shaped line and the number of permutations of lines connecting the few points, something like that. You remember that?