Archive for 2009


The mystery of number 23

I’ve been literally forced to write this post. My hands have been dragged to the keyboard and my mind has been controlled. Never would a person who claims to be rational, scientific and no-nonsense would write on something as unscientific as why number 23 is special. For the uninformed, there is something deeply mysterious, extremely special and wonderfully interesting about 23. The phenomenon is called as 23 Enigma.

When I first heard about it, it was yet another overhyped meme for me. But slowly, and definitely steadily, I started identifying 23 in everyday phenomenon around me. Now, not a day passes without me seeing 23 somewhere around me. And it is not just me, by showing examples, I have convinced several of my friends about this odd belief of mine.

That said, internally, I debate endlessly with myself whether what I observe is just an artifact of confirmation bias where what one observes is typically due to what one wants to observe. Soon I plan to start a journal where I would try to statistically see if observation of 23 is more frequent than what is expected by random chance.

Do you observe 23 as well? I would be really interested in listening to your story.

BTW: Would you be shocked if I predict that within next two days you are certain to observe number 23 somewhere around you :)

April Fools Pranks on the Internet

I randomly stumbled across many interesting April Fools pranks on the Internet today. So, I thought of sharing them here:

Enjoy! I will try to add more as I come across them. Do let me know if you any interesting pranks.

So, you have a million dollars. Now what?

Suppose you win a lottery or get a successful exit for your startup. You suddenly find yourself with a million dollars to play with. It is like what you always dreamt for has become a reality. At this point, what would you do? What would your future look like?

It is hard to imagine not having fun with a million dollars in bank. But that is precisely what I (sadly) concluded after thinking deeply over this issue. There are multiple reasons why I think a million dollars won’t bring me the freedom, happiness and euphoria that I had thought were main motivators for me.

One of most the reasons concern with the social circle. Even if you get lots of money to spend, with whom would you go shopping? Who would be able to afford accompanying you to your always-dreamt-of-visiting Morocco? Is your friend equally fond of I-would-go-bungee-jumping-one-day? Chances are that most of your friends or relative would not have time to be there with you all the time, would not be able to afford hanging out with you while you are indulging, and would politely decline your sponsorship. So, my friend, what would you do with your million dollars?

How many movies per week can you watch? How many books can you read? Can you eat at fine-dine restaurants three times a day for a whole month? To be realistic, I find it hard having a tremendous amount of joy from a million dollars. I don’t think I would be more happy then than I am at this moment. Maybe the kick of having money far more strong than the actual pleasure in it.

Luck, Randomness and Success

Is success random? How important do you think luck is for achieving success? Now, you may be a self-hero, a believer in hard-work-brings-success philosophy; every morning you may look forward to reading stories of successful people who were certain of their successes from the start. To achieve success, you may even be sufficiently motivated to learn new skills, network madly, and work endlessly. After all, if you do all the things right, success will kiss you. See, you are feeling great already! Now that you are all set to achieve success, let us sit in a time machine and head forward three years in future.

You have grown a little older. But, sadly, nothing really has changed. All what has changed is that your book shelf has ten more books of successful people and twenty more of self-help books. Looking back you deeply regret working like hell for the last three years. You feel pangs of guilt whose source cannot be traced. You feel cheated. You feel like burning all the fake books your book shelf is having.

Whoa! Wait, we almost forgot about your twin brother. Let us ask how he feels now on recently selling his company for $25 MM. He tells you that he feels great and because he did yada-yada in life, he was able to get a profitable exit. You know deep in your heart that you are way smarter than him, had more connections, worked twice as hard. But he is successful, you are not. So, his success must be a fluke. Right?

Right. Repeat after me what you just said. Success is a fluke. Success is a fluke.

So, why obsess over it? Why loathe success of your brother? Why read self-help and startup stories books? Why burn your soul daily for not achieving success? Why regret being so smart?

Today, you have still three years before realizing success is a fluke, so make most of it.

But then you ask, why does success look so methodical. Why do successful people say that what they learnt, where they went for studies, whom they met at a party, etc. played a major role in their success. Why do the dots look connected as if there is indeed a recipe for success? This is because dots are in fact connected. Even for great failures, dots look connected. Heck, even for normal lives, dots look connected. Dots are made to be connected.

Apart from the usual stuff, there are a million other things which influence success. After a minimum quantity of basic ingredients: networking, hard work, skills, you should leave it on randomness to get you success. Don’t obsess over it. Just be ready with basic ingredients and expose yourself to randomness. Given enough time, success should follow.

And don’t fret over your twin brother’s success, after all he is your brother. Moreover, if he got lucky in three years, you may also get lucky in another two, ten or maybe thirty years. if you never get success, don’t blame it on you, blame it on lady luck!

Wonders of Probability

Lately, I have been reading a lot about Probability, Statistics and Machine Learning. These subjects always involve a kind of awe that one can only experience on understanding the concepts. I’ve developed tremendous respect for the people who understand probability, (Bayesian) statistics, learning theory and other related subjects.

I write this post to pose some probability (pseudo) paradoxes and problems to you. Even though these paradoxes are extremely simple to understand and comprehend, but their solution is completely counter-intuitive. I’ll admit that for some of the paradoxes/problems, even after seeing the solution I wasn’t able to figure out what exactly is going on. So, without further blabber here we go with the paradoxes:

First one is called the Monty Hall Problem. Its statement goes something like this:

Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?

Think a lot about the problem. The solution is not as simple as it seems!

While on the surface it seems that it should not matter whether you switch the door or not, switching actually turns out to be advantegous for you. In fact, with the second door you will have 66% of chance of winning while the existing door will give you 33% chance of winning. See the wikipedia article for explanation.

The second one is even more interesting. It is taken from the document titled Nuances of Probability. The paradox goes something like this:

My neighbor has two children. Assuming that the gender of a child is like a coin flip, it is most likely, a priori, that my neighbor has one boy and one girl, with probability 1/2. The other possibilities—two boys or two girls—have probabilities 1/4 and 1/4.

Suppose I ask him whether he has any boys, and he says yes. What is the probability that one child is a girl? By the above reasoning, it is twice as likely for him to have one boy and one girl than two boys, so the odds are 2:1 which means the probability is 2/3. Bayes’ rule will give the same result.

Suppose instead that I happen to see one of his children run by, and it is a boy. What is the probability that the other child is a girl?

Again, you need to really understand what is being asked in the question. To give you some guidance, here is what solution looks like (don’t worry, even after seeing the solution it would be hard to believe that it is indeed the solution).

Observing the outcome of one coin has no affect on the other, so the answer should be 1/2. In fact that is what Bayes’ rule says in this case. If you don’t believe this, draw a tree describing the possible states of the world and the possible observations, along with the probabilities of each leaf. Condition on the event observed by setting all contradictory leaf probabilities to zero and renormalizing the nonzero leaves. The two cases have two different trees and thus two different answers.

So, it is an apparent paradox! And I don’t know how it is resolved. If you get to understand this, let me know in the comments.

Another simple problem relating to Bayes’ theorem which people usually get wrong is as follows:

1% of women at age forty who participate in routine screening have breast cancer. 80% of women with breast cancer will get positive mammographies. 9.6% of women without breast cancer will also get positive mammographies. A woman in this age group had a positive mammography in a routine screening. What is the probability that she actually has breast cancer?

What do you think the answer is? Don’t worry if you get the answer wrong. You know what, even after considering how critical will a misdiagnosis of breast cancer turn out to be, only 15% doctors get it right. That is indeed scary! This also hints at the importance of understanding and appreciating the depth of probability.

If you apply Bayes’ theorem carefully, the answer will turn out to be 7.8%. And just for your information, most doctors estimate the chance of having cancer to be between 70% and 80%.

The above were some of the interesting paradoxes cropping up due to interpretation of probability and inability of people to apply Bayes’ theorem mentally. If you are game for even more probability paradoxes and problems, head to this wikipedia article.