Regarding the Ellsberg paradox, seen last week, this is what our “featured user” Manuel Amorós says:
We could think that the situation of the yellow and black balls is symmetrical and, therefore, the probability of drawing yellow is the same as that of drawing black, which together with the fact that the probability of drawing red is 1/3, leads us to that the three probabilities are equal. In this case it is indifferent to choose A or B. And something similar happens in the second choice, where the probability of winning is 2/3 in both cases.
I understand that the proposed situation would be equivalent to the following: we have 61 ballot boxes. In all of them there are 30 red balls. In the first urn there are also 0 black balls and 60 yellow balls; in the second, 1 white and 59 yellow, etc. An urn is chosen at random and a ball is drawn. Modeled in this way, it is seen that the three colors have the same probability and the mystery disappears.
(I’m sure my astute readers notice some parallels with the problem of Abdul’s black and white balls, in The Condemned Man and the Urns, dedicated to George Gamow).
Yule-Simpson Effect
Decision theory, linked to the calculation of probabilities, gives rise to other interesting paradoxes. Let’s look at an example:
You have to pass three consecutive tests. In the first test, there are two bags, A1 and B1, in which there are, respectively, 6 white balls and 5 black balls, and 4 white balls and 3 black balls. You have to choose one of the two bags (whose contents you know) and draw a ball at random. If it is black, you are eliminated, and if it is white you go to the next test.
In the second test, there are two other bags, A2 and B2, in which there are, respectively, 3 white balls and 6 black balls, and 5 white balls and 9 black balls. You choose a bag, put your hand in, and if you pull out a white ball you move on to the next test.
In the third test, you have to choose between bag A, in which the balls of A1 and A2 have been put together, and bag B, in which the balls of B1 and B2 have been put together (after reintroducing the balls taken previously). , and, once again, you win if you draw a white ball. Which bag would you choose in each case? Are you surprised by your own answer? Because?
This experiment illustrates a decision theory paradox, well known to statisticians, called “Simpson’s paradox” in honor of the recently deceased British mathematician Edward H. Simpson, famous for being the cryptanalyst who deciphered the messages of the Italian Navy during the Second World War. In fact, the paradox had already been mentioned in 1903 by the Scottish statistician George Udny Yule, and, on the other hand, some do not consider it a paradox per se, which is why they prefer to call it the “Yule-Simpson effect.”
In addition to the experiment described, there are numerous real situations that illustrate this misleading effect. One of the best-known examples is the lawsuit against the University of California, Berkeley, in 1973, for discrimination against women. The graduate admissions results seemed clearly discriminatory:
Men: 8,442 applications, 3,714 admissions (44%)
Women: 4,321 applications, 1,512 admissions (35%)
However, a detailed analysis of the data showed that there had been no such discrimination and the lawsuit was dismissed. How is it possible? Can you think of any possible breakdown of the data that would lead to this conclusion? And how would you define the Yule-Simpson effect in view of the balls experiment and this real case of false discrimination? (Hint: The Yule-Simpson effect is also known as the “amalgamation paradox” and the “reversal paradox.”)
You can follow SUBJECT in Facebook, x and instagramor sign up here to receive our weekly newsletter.
Subscribe to continue reading
Read without limits
_
#Simpsons #paradox