In our experiment last week, it is evident that it is advisable to choose bags B1 and B2, since in them the probability of drawing a white ball is, respectively, 4/7 and 5/14, while in bags A1 and A2 it is 6/11 and 3/9 = 1/3, slightly less in both cases. However, when putting the balls together in bags A and B, in the first the probability of drawing a white ball is 9/20 and in the second 9/21, so now, and counterintuitively, it is convenient to choose the bag A. That is why the Yule-Simpson effect is also called the “amalgamation paradox”, because when two favorable options are combined the sum becomes unfavorable.
My regular readers may remember the riddle of the paradoxical chocolates, published exactly eight years ago in the installment “Poisoned Chocolates” (11 6 2015) and which at the time achieved some popularity on social networks; Well, although we didn’t name it then, it is a clear example of the Yule-Simpson effect.
Regarding the case of alleged discrimination at the University of California, a detailed examination of the applications revealed that, in general, women had applied for more difficult graduate courses, in which the percentage of admissions was lower for both men and women. women, which explained the apparently discriminatory result. The Yule-Simpson effect works in both directions: by putting together, as in the case of balls and chocolates, and by breaking down, as in the case of false discrimination (which is why it is also called the “reversal paradox”).
The drunk photons
As we have seen on several occasions, the calculation of probabilities and random processes frequently give rise to paradoxical or counterintuitive results. And one of the most surprising and little-known takes place inside our Sun.
If we go back in time a little less than in the case of chocolates, five years ago, in the installment “The Drunkard’s Walk” (14 12 2018), we saw that the erratic walk of a drunk is usually used as a model of the more varied random processes. Let’s imagine the well-known drunk clinging to a lamppost who suddenly decides to walk and takes one meter steps (to simplify, we will choose a long-legged drinker). If he walked in a straight line, after taking 100 steps he would have moved 100 meters away from the streetlight; but if after each step it changes the direction of its movement at random, as is typical of its lamentable state, it is easy to demonstrate that it is most likely that it will only move away about 10 meters: after n unit movements, a random mobile on a plane moves moves away from the starting point, on average, √n units.
It takes sunlight about 8 minutes to travel the 150 million kilometers from the Earth to the Sun; but the photons that form inside our star take a little longer to go out into space. If a photon starting from the center of the Sun crossed it in a straight line, it would take just over two seconds to travel the 700,000 kilometers of the solar radius; but the photon continually collides with particles that deflect it, and it is like a drunk taking random centimeter steps; Therefore, it takes a long time for the tightly packed photons to leave the Sun before they can launch themselves across space at 300,000 kilometers per second. How long does it take, on average, for “drunk” photons to reach the solar surface?
(Without intending to invade the interdisciplinary terrain of Montero Glez, I will point out that the title of this entry is a tribute to the excellent collection of poems by Julio Llamazares The slowness of the oxen. Like photons, oxen can be very fast – up to 60 kilometers per hour – or move peacefully slowly.
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