Regarding the Ellsberg paradox, which we saw last week, this is what our “super user” Manuel Amoros has to say:

We can think that the positions of the yellow and black balls are the same, so the probability of drawing yellow is the same as the probability of drawing black, which brings us to the fact that the probability of drawing red is 1/3 so the three probabilities are equal. In this case it does not matter whether A or B is chosen. Something similar happens in the second…

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Regarding the Ellsberg paradox, which we saw last week, this is what our “super user” Manuel Amoros has to say:

We can think that the positions of the yellow and black balls are the same, so the probability of drawing yellow is the same as the probability of drawing black, which brings us to the fact that the probability of drawing red is 1/3 so the three probabilities are equal. In this case, it does not matter whether you choose A or B. Something similar happens with 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. There are 30 red balls in each of them. In the first jar there are also 0 black balls and 60 yellow balls; And in the second 1 white, 59 yellow, etc. The urn is chosen at random and the ball is drawn. As such, it turns out that the three colors have the same probability and the ambiguity disappears.

(I’m sure my astute readers have noticed some similarities to the problem of Abdul’s black and white balls, in The Condemned Man and the Urns, assigned to George Gamow.)

### The 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 tests in a row. In the first test, there are two bags, A1 and B1, which contain respectively 6 white balls and 5 black balls, and 4 white balls and 3 black balls. You choose one of two bags (the contents of which you know) and draw a ball at random. If it’s black, you’ll be eliminated, if it’s white, you’ll move on to the next test.

In the second test, there are two more bags, A2 and B2, which contain respectively 3 white balls and 6 black balls, and 5 white balls and 9 black balls. You choose a bag, put your hand in it, and if you draw a white ball, you move on to the next test.

In the third test, you have to choose between bag A, in which balls A1 and A2 are grouped together, and bag B, in which balls B1 and B2 are grouped together (after reinserting the previously taken balls). And again, you win. If you draw a white ball. Which bag will you choose in each case? Were you surprised by your own answer? because?

This experiment illustrates the decision theory paradox, known to statisticians, called the “Simpson’s Paradox” in honor of the recently deceased British mathematician Edward Simpson, famous as the codebreaker who deciphered Italian Navy messages during World War II. In fact, the paradox was previously mentioned in 1903 by the Scottish statistician George Udney Yule. On the other hand, some do not consider it a paradox in itself, which is why they prefer to call it the “Yule-Simpson effect.” “.

In addition to the described experiment, there are several real situations that illustrate this misleading effect. One of the most famous examples is the lawsuit filed against the University of California, Berkeley, in 1973, for discrimination against women. Postgraduate admissions results appeared clearly discriminatory:

Men: 8,442 applications, 3,714 acceptances (44%)

Women: 4,321 applications, 1,512 accepted (35%)

However, detailed analysis of the data showed that there was no such discrimination and the claim was dismissed. How is that possible? Can you think of any possible detail of the data that might lead to this conclusion? How do you define the Yule-Simpson effect in light of the balls experiment and this real case of false discrimination? (Hint: The Yule-Simpson effect is also known as the “fusion paradox” and the “reversal paradox.”)

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