Grown-ups love figures. When you tell them that you’ve made a new friend, they never ask you any questions about essential matters. They never say to you, “What does his voice sound like? What games does he love best? Does he collect butterflies?” Instead, they demand: “How old is he? How many brothers has he? How much does he weigh? How much money does his father make?” Only from these figures do they think they have learned anything about him.
-The Little Prince
Our first exams in this first semester at ISI have finally arrived. One of the subjects of the course, and of the more interesting ones I might add, is ‘Probability and Stochastic Processes’. This was also the subject of our first exam. I have been pouring over notes and hundreds of pages of text in preparation of this exam. The intricacies in some of the ideas reminded me of these lines taken from ‘The Little Prince’ and quoted by David Freedman in his book on Statistics. Are such matters as figures and charts always dry and boring? In our first class, the professor of the same subject also remarked, “Maybe our insistence on numbers is the limitation of man”.
‘The Little Prince’ is a famous classic written by Saint Antoine du Exupery. Although it was first written as a children’s book, it has been enjoyed over the years by adults alike as a heart-warming story on friendship, growing-up and the curious fascinations of man. One of the oft used coinages in the book is the phrase ‘matters of consequence’. The Little Prince finds, at one point, a man aimlessly counting stars to the extent that he reaches ‘five hundred and one million’. This man then ironically refers to his counting as a matter of consequence.
In these days leading up to our exams, I began to wonder whether factuality could be made interesting in the context of some intricacies I came across in problems of Probability Theory. Are there questions that I can ask that are not as dry as the character’s occupation of counting in The Little Prince?
Consider the following questions. How many random people would I need to collect in a room, such that there would be half a chance that at least two of them share their birthdays? The answer to this is a surprisingly low number. Let’s look at a different question. How many people would I need to catch hold of and ask birthdays of to have half a chance of finding someone who shares my birthday? It is also interesting to note that this number is different from the first one. Louis CK, the famous stand-up comedian, made an interesting remark in one of his live shows. He said that there were enough people attending his show for there to be a fair chance that a few of them would die in the coming year. Ouch! Let’s pose that as a question on chance. Assume a population with a certain death rate. What is the number of people I would have to randomly collect in a room, such that there would be half a chance that someone would die tomorrow?
Enough with birthdays and death days. Consider coin tosses. Say, I keep tossing a coin and keep getting heads for a hundred tosses. I only know that it is equally likely that my coin is anything between a perfectly rigged coin to a perfectly fair coin. What is the chance that I win if I bet on the next toss to be heads? Let’s stretch this slightly further. Let us assume that when the universe was born, the probability that the sun would rise over planet earth each day was made to depend on a coin toss. We only know that this coin was likely to be anything between a perfectly fair to a perfectly unfair coin (God is playing a cruel game). What is the probability that the sun would rise tomorrow, given that it has been rising each day over all these years since the birth of the universe?
Let’s give some rest to our poor coin and move on to other questions. The Monty Hall Problem is a famous example on how sometimes the idea of chance can stump intuition. Here’s a different example in a similar vein. Suppose there are three suspects in jail who have been cleared of any wrongdoing and are all going to be released soon. It is announced that two of the three would be released the next day but these three don’t know exactly which two of them will be released. One of them decides to go ask the guard which one of the other two prisoners is going to be released. But then he thinks the following to himself, “After I have asked, I have only a one in two chance that I’m the second guy to be released. But before I ask, there is a two in three chance that I will be one of the guys to be released. So should I rather not ask?”
Like one of our professors quips so often, ‘Remember that there’s a chance model somewhere in the background.’ Somebody somewhere is flipping a coin. Maybe someday a story will be written on a travelling mathematician who visits magical planets to ask questions on chances of consequence. It’s a long shot that it would be anything as beautiful as the masterpiece that is the ‘The Little Prince’. But I wager it wouldn’t be as bad as dry pointless counting.