Probability and Expectation
Name: Tuan C.
Given two boxes, and told that one contains double the
amount of money of the other. When one box is chosen at random and
opened, there were 200 dollars. Most would reason that the other box will
contain either 100 or 400, and therefore the expectation is 250, and
hence it is superior to the first box.
Since the first box was chosen at random, is there a fallacy in applying
probability considerations here? Thanks.
To use the expectation of 250 as a definition of what is in the box is the
fallacy. Consider the two options, as compared to taking the first box. If
you choose the second rather than the first, you might lose 100 dollars. On
the other hand, you might gain 200 dollars. By choosing the other box, you
have a 50% chance of losing $100 and a 50% chance of gaining $200. If the
same option with the first box containing $200 comes up 10,000 times, always
choosing the first box results in a gain of $2,000,000. Always choosing the
second box results in a gain of $2,500,000. A person who likes risks will
choose the second box because he stands to win more than he can possibly
Dr. Ken Mellendorf
Illinois Central College
This apparent paradox is resolved by: 1. recognizing that the
information "one contains double the amount of money of the other." removes
the condition that amounts of money are randomly distributed. and 2. the
proper statement of the "expectation value" is that "There is a 50% chance
that the second box will contain either $100 or $400."
That there is a misapplication of the term "expectation value" consider
the analogous "experiment". A coin has a 50% chance of landing "heads" or
"tails". If the coin is tossed again there is still a 50% chance of the coin
landing "heads" or "tails". The events must also be independent, which is
not the case if you know the amounts in the second box.
Finally, restate the problem, "The second box contains either $1 or
$999." You have a 50/50 chance of getting either $1 or $999. It is the
"boxes" you are averaging, not the "amounts" in the boxes. That has no
relevance regarding the probability of choosing the second box.
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Update: June 2012