If Abdul, our condemned man from last week, had been able to distribute the white and black balls in any number of urns, he would have been almost certain of being saved, putting in 50 urns a white ball in each one, and the 50 black balls all in another urn; Thus, his probability of drawing a white ball would have been 50/51. He only has two urns, but the extreme case we have just seen (many problems are clarified by taking the situation to the limit) suggests the optimal strategy: put a white ball in one urn and the remaining 99 in the other; In this way, he has a 50% chance of choosing the urn with the white ball, and if he chooses the other he has a 49/99 chance of getting a white ball: overall, he has almost a 75% chance of being left. free.
The urn (or bag) problems with black and white balls are a classic of probability calculation, and can give rise to interesting paradoxes, such as that of the Bertrand box, which we have dealt with on more than one occasion.
Paradoxical decisions
But drawing balls at random not only lends itself to illustrating probabilistic paradoxes, but also paradoxes of decision theory (which studies the behavior and psychological processes of people who have to make decisions). One of the best known is the Ellsberg paradox (named after having been formulated by the recently deceased American analyst Daniel Ellsberg), which shows that when faced with a choice between two options based on incomplete information, most people choose that whose probability is known, even against the principle of independence of the decision theory (which we will deal with at another time).
In 1961, Ellsberg performed the following experiment:
In an urn there were 90 balls, 30 red and the rest yellow or black in an unknown proportion, and a series of people were presented with the following option:
TO. If you draw a red ball you win a certain amount of money, if it is black or yellow you lose.
b. If you draw a yellow ball you win, if it is red or black you lose.
Most subjects chose option A.
Immediately afterwards, two other options were presented to the same subjects and with the same balls:
c. If you draw a red or black ball you win, if it is yellow you lose.
d. If you draw a yellow or black ball you win, if it is red you lose.
In this case, the majority of the subjects chose option D. What would you have chosen in both cases? Again: where is the paradox?
The most dangerous man
But the phrase “Ellsberg’s paradox” could also be understood in another way. In 1971, Daniel Ellsberg, while working at the Rand Corporation, leaked to the New York Times the so-called “Pentagon Papers”, top secret documents on the decisions of the United States Government in relation to the Vietnam War, for which he was persecuted by the Nixon administration and called “the most dangerous man in the United States.” Paradoxically, the truly dangerous men branded the pacifist who had unmasked them as dangerous.
Ellsberg’s denunciation has been made into a film on two occasions: The Pentagon Papers (1993), by Rod Holcomb, and ThePost (2017), by Steven Spielberg.
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