If Abdul, our convict from last week, had been able to distribute the white and black balls to as many urns as he wanted, he would almost certainly have been saved if he had put one white ball in every 50 urns, and the 50 black ones balls, all in a different urn; So your probability of drawing a white ball would have been 50/51. You only have two urns, but the extreme case we just saw (many problems are solved by pushing the situation to the extreme) suggests the optimal strategy: put a white ball in one urn and the remaining 99 into the other; This way, you have a 50 percent chance of choosing the urn with the white ball, and if you choose the other, you have a 49/99 chance of getting a white ball: in total, you have almost a 75 -percent chance of being free.
Urn (or bag) problems with black and white balls are a classic in probability calculations and can lead to interesting paradoxes, such as that of the Bertrand box, which we have dealt with more than once.
Paradoxical decisions
But randomly drawing balls is useful for illustrating not only probabilistic paradoxes, but also paradoxes of decision theory (which studies the behavior and psychological processes of people who must make decisions). One of the best known is the Ellsberg paradox (named after the formulation of the late American analyst Daniel Ellsberg), which shows that when most people have to choose between two options based on incomplete information, they choose the one whose probability is known . even against the independence principle of decision theory (which we will deal with another time).
More information
In 1961, Ellsberg conducted the following experiment:
In an urn were 90 balls, 30 red and the rest yellow or black, in an unknown ratio, and a number 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, the same subjects and with the same balls were presented with two further options:
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 subjects chose option D. What would you have chosen in both cases? Again, where is the paradox?
Paul Giamatti alongside James Spader as Daniel Ellsberg in The Pentagon Papers (1993) Photo: MPTV.net (Cordon Press)
The most dangerous man
The expression “Ellsberg’s paradox” could also be understood differently. In 1971, while working at the Rand Corporation, Daniel Ellsberg leaked to the New York Times the so-called “Pentagon Papers,” top-secret documents about the U.S. government’s decisions related to the Vietnam War, over which it was given responsibility persecuted by the Nixon administration and described as “the most dangerous man in America.” Paradoxically, the truly dangerous men branded the pacifist who exposed them as dangerous.
Ellsberg’s denunciation has been made into films twice: “The Pentagon Papers” (1993) by Rod Holcomb and “The Post” (2017) by Steven Spielberg.
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