Had a nice though-provoking evening with Shibin. Thought would share it to see what other think about this matter. Before I start, I apologize for using ‘death’ as a medium of analysis. This post is purely academic and I seek forgiveness if it offends anyone’s feelings.
Most of the experiences we have in our lives are everyday experiences that we expect to happen: We expect for the sun to rise the next day. We expect for the signal to turn from red to green. We expect not to get in to an accident when we walk out of the house. We expect our house not to burn down. We expect ‘u’ to follow ‘q’. We expect not to trip and fall down. We expect not to meet our childhood friend, whom we have not met for past 10 years. When most of these expected events happen, or the unexpected events dont happen, we are not curious to know why did it happen or not happen, but sometimes unexpected events happen: Even if we dont expect to trip and fall, we might trip and fall. Even if we dont expect to meet our childhood friend, we might just bump into him one fine day. We are amused by these events, and although they are unexpected, our intuition knows that these events happen once in a period of time and we still keep on living our lives without much thought about why did these events happen. But some other times rare events happen. And that puzzles us. What if we trip and fall twice on the same day. And then twice again? Did I do something wrong yesterday? What if I meet a friend of mine in a supermarket, and her again in a movie hall? Does god has some plans for us. What if someone’s elder brother dies at the age of 23, and he himself meets a fatal accident at that very age of 23? Is the family cursed? Was the entry of someone in the family the reason for that curse?
These rare events confuses us. On one hand our intuition tells us that these events are not supposed to happen. But they happen to us and then we wonder, is there some meaning behind them? It feels too much of a coincidence to happen. Shibin felt that two brothers dying at the same age of 23, at different independent times, is a rare event to happen. I felt the same. I mean what are the odds of this happening? I mean literally what are the odds? My intuition tells me that this is a rare event, but how rare of an event this is? If I know of 100 families with two brothers, do I expect that one of this family will meet with such a sorrow fate? We thought of computing this odd. Not sure if the math and reasoning is right, but we were pretty surprised by the outcome.
We want to compute the probability that two brothers die at the same age. Be that 23 or 32 or any other age. To me the shock comes from the fact that they died at the same age. Now to answer that question, for simplicity, lets assume that the world statistic for the probability of death at different ages is uniform. That is, assuming a human lives from anywhere between 1-100 years, the probability of dying at any age is 1/100 = 0.01. This is a simplification as the actual distribution would me more close to a Gaussian distribution (a bell curve). For example, the distribution for US deaths in 2006 is
But for the sake of the discussion, let it be 0.01. Now given this, can we compute the probability? Our first attempt gave us this
Prob(brothers dying at same age) = Prob(brothers dying at age 1) + Prob(brothers dying at age 2) + … + Prob(borther dying at age 100).
And to compute any one of the probability we have
Prob(brothers dying at age x) = Prob(Son1 dies at x, Son2 dies at x) = Prob(Son1 dies at x) Prob(Son2 dies at x) = 0.0001
Prob(brothers dying at same age) = 0.01 (Infact its 0.02 based on the actual probability graph above).
Which means that given 100 families we would expect 1-2 families on average to meet with this sorrow fate. Not too low a probability to make us look into any meaning behind the event?
Infact since we are on this topic, I get reminded of old hindi movies where the bride was considered a bad omen, if the husband dies in a couple of years after the marriage. Looking at the plot above, 0.5% of people die between the age of 25-30. Assuming that all were married in the preceeding 5 years and that the plot holds for India too, with a population of 1billion, about 5 million people die between 25-30. Surely enough all these deaths have a meaning.
30 Jan 2010 UPDATE: Thanks to Anupam Dubey, I am motivated to write this update. He went thorough the math and something did not seem right. Actually I too went over the math again later, and found how statistics have to be handelled better. The math is right. Its what we are computing is wrong. Restricting to the actual probability graph, let us compute another probability:
Prob(brothers dying at same ‘young’ age ) = Prob(brothers dying at age 1) + Prob(brothers dying at age 2) + … + Prob(borther dying at age 50) [where we define ‘young’ to be less than 50]
This probability is actually 0.0001 (in contrast to 0.02 for two people dying at the same age). This is a little more in line with the intuition, as now given that you know 10,000 families, you are likely to find 1 family where two people died at the same age between 0-50. Furthermore, as per the previous calculations you are likely to find 200 families where two people died at the same age between 0-100.