In our experiment last week, it is clear that it is advisable to choose bags B1 and B2, since in them the probability of getting a cue ball is, respectively, 4/7 and 5/14, while in bags A1 and A2 it is 6/11 and 3/9 = 1/3, slightly lower in both cases. However, when putting the balls together in bags A and B, in the first the probability of drawing a cue ball is 9/20 and in the second 9/21, so now, and counterintuitively, it is convenient to choose bag A. That is why the Yule-Simpson effect is also called the "amalgamation paradox", because when two favorable options are put together, the sum becomes unfavorable.
My regular readers may remember the riddle of paradoxical chocolates, published exactly eight years ago in the issue "Poisoned Chocolates" (6 11 2015) and which once achieved some popularity on social networks; Well, even if we didn't name it then, it's a clear example of the Yule-Simpson effect.
With regard to the case of alleged discrimination at the University of California, a detailed examination of the applications found that, in general, women had applied for more difficult postgraduate courses, where the percentage of admissions was lower for both men and 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 (this is why it is also called the "paradox of reversal").
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Illustrious Unknowns
The Drunken Photons
As we have seen on several occasions, probability calculation and random processes often lead 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 drunkard is usually used as a model for the most varied random processes. Let's imagine the well-known drunk clinging to a lamppost who, suddenly, decides to start walking and starts taking steps of one meter (for simplicity, we will choose a long-legged drinker). If he were walking in a straight line, after taking 100 steps he would have moved 100 meters away from the lamppost; But if, after each step, the direction of its march changes randomly, as is typical of its deplorable state, it is easy to show that it is most likely to move away only about 10 meters: after n unit displacements, a random mobile on a plane moves away from the starting point, on average, √n units.
It takes about 8 minutes for sunlight to travel the 150 million kilometers that the Earth is from the Sun; But the photons that form inside our star take a little longer to get out into space. If a photon starting from the center of the Sun were to pass through it in a straight line, it would take a little more than two seconds to travel the 700,000 kilometers of the Sun's radius; but the photon continually collides with particles that deflect it, and is like a drunkard taking random steps of a centimeter; therefore, the squeezed photons take a long time to leave the Sun before they can launch themselves through space at 300,000 kilometers per second. How long does it take, on average, for "drunk" photons to reach the solar surface?
(Without wishing to invade the interdisciplinary terrain of Montero Glez, I will point out that the title of this post is a tribute to Julio Llamazares' excellent collection of poems La lentod de los bueyes. Like photons, oxen can be very fast – up to 60 kilometers per hour – or move peacefully slowly.)
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