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Paradoxes From Probability Theory:
Mutual Exclusivity
John D. Norton
Department of History and Philosophy of Science
University of Pittsburgh
http://www.pitt.edu/~jdnorton
For a compact reminder of probability theory, see Probability Theory Refresher.
The probability theory provides a quantitative calculus for dealing with chance and uncertainty. It is one of the most successful analytic tools available to us and is often called upon to correct misconceptions about chance. These corrections have an air of paradox to them, since they give results that are, at least initially, quite unexpected. They have a strong presence in the paradox literature. They are not paradoxes of the type that reveals a contradiction in our presupposition. Rather they are paradoxes in the sense that they present us with unexpected results. This chapter presents a brief sampling of them.
Since the paradoxes arise through corrections FROM probability theory TO common misconceptions, they belong in the "from" chapters. Subsequent chapters will investigate a reversed problem. While the probability calculus can be applied profitably to a very large array of problems, there are some circumstances in which it fails. These are presented as paradoxes FOR probability theory.
The paradoxes from probability theory can be categorized loosely into those that arise from improper assessments of mutual exclusivity, from improper assessments of probabilistic independence and from improper assessments of the import of expectations. Examples of each are given in this chapter and the following two chapters.
The commonality in these paradoxes is that they involve a failure to consider all the mutually exclusive outcomes that comprise the outcome space, in a space in which all the most specific, mutually exclusive outcomes have the same probability. Rather, two or more mutually exclusive cases are treated as one case and thus their probabilities are underestimated.
This paradox, already described in the Budget of Paradoxes involves just this mistake concerning mutually exclusive outcomes. Here is the paradox again:
At a carnival sideshow, you are invited to play the following game. There are three cards. One is blue on both sides. One is red on both sides. One is blue on one side and red on the other.
To play the game, you are allowed to shuffle and flip the cards without looking until they could be in any order with any side up.
A card is drawn and it is red. You are offered an even odds bet that the other side of the card is blue.
It seems a fair bet. The color of the other side of the card is either red or blue, so each has (it would seem) an equal chance. That is:
(??) The probability that the color of the other side of the card is blue is 1/2. (??)
This is incorrect and overestimates the chance that the other side is blue, which then inclines us to an unfavorable wager. The ease with which we fall into this error makes it easy for the swindle to occur.
The error in the analysis is that the set of mutually exclusive outcomes has not been assessed correctly. When the card is drawn, there are three mutually exclusive outcomes possible:
The card is red-blue and the red side is uppermost.
The card is red-red and side-1 is uppermost.
The card is red-red and side-2 is uppermost.
If we imagine that the two sides of the red-red card are explicitly numbered, the three possible, mutually exclusive outcomes are shown as:
The shuffling and flipping described above ensures that each of these outcomes has equal probability. In two of three outcomes, the other side of the card is red; and in only one is the other side blue. Since these three outcomes are equally probable, it follows that
The probability that the color of the other
side of the card is blue is 1/3.
Another common formulation of what is essentially the same paradox concerns a two-child family. The essential background assumptions are that:
probability of boy P(B) = probability of girl P(G) = 1/2
and that the probabilities of the gender are independent of the genders of the other children in the family.
Until we have more information that restricts the possibilities, the outcome space is comprised of four mutually exclusive outcomes:
GG,
GB, BG, BB
where "GG" = "first child is a girl; second child is a girl" etc.
This idealized set of assumptions could equally be realized with independent coin tosses.
??
We are asked two question:
"one": One of the children is girl, G. What is the probability that the other is girl, G?
"first": The first child is a girl, G. What is the probability that the second is a girl, G?
It is easy to assume that both questions have the same answer. For in both cases, the other child might be a boy, B, or a girl, G. These two cases then suggest a probability of 1/2 for each. That is:
(??) In both "one" and "first," the probability of a girl, G, is 1/2. (??)
Once again, this conclusion is mistaken. The two questions each lead to different sets of possible, mutually exclusive outcomes and thus different reduced outcome spaces.
In the case of "one," we know that one of the children is a girl. We are not told which is the girl. It could be either the first or the second child. Hence there are three remaining mutually exclusive outcomes possible:
GG,
GB, BG
Each has equal probability. In two of these outcomes, the other child is a boy; and in only one is the other child a girl. So we arrive at the correct result:
In "one," the probability of a girl, G, is 1/3.
The case of "first" leads to a different set of mutually exclusive outcomes that are still possible. They are:
GG, GB
Each has equal probability and in only one of them is the other, second child a girl. So we have a different result from "one":
In "first," the probability of a girl, G, is 1/2.
A more striking difference between questions like "one" and "first" appears if we increase the size of the family.
??????????
The revised questions for a ten-child family are:
"nine": nine of the children are girls, G. What is the probability that the other is girl, G?
"first nine": The first nine children are girls, G. What is the probability that the tenth is a girl, G?
As before, we should resist the temptation to say that in each case the probability of girl, G, is the same: 1/2. The two cases are seen to be very different if we enumerate the mutually exclusive possibilities each allows.
In the case of "nine," there are eleven mutually exclusive possibilities that form the reduced outcome space. Using the notation above, they are:
GGGGGGGGGG,
GGGGGGGGGB,
GGGGGGGGBG,
GGGGGGGBGG,
GGGGGGBGGG,
GGGGGBGGGG,
GGGGBGGGGG,
GGGBGGGGGG,
GGBGGGGGGG,
GBGGGGGGGG,
BGGGGGGGGG
In only one of these eleven mutually exclusive outcomes is the remaining child a girl. Since these are equally probable, we have:
In "nine," the probability of a girl, G, is 1/11.
The case of "first nine" is different. There are only two remaining, mutually exclusive outcomes possible:
GGGGGGGGGG,
GGGGGGGGGB
In one of these two cases, the tenth child is a girl. Since these mutually exclusive outcomes are equally probable, we have:
In "first nine," the probability of a girl, G, is 1/2.
We might try to conceive these last paradoxes as arising from errors in judgments of independence. It may seem, for example, that the outcome of nine children being girls and the outcome of the remaining child being a girl are two dependent outcomes. The error is to treat them as independent. This is a weaker diagnosis. The difficulty is that the second outcome in this analysis--the other child is a girl--is not an outcome well-defined prior to specification of the first outcome (that nine children are girls). That imprecision is the deeper cause of the problem and better remedied by a more careful description of the mutually exclusive outcomes of the reduced outcome space.
How is it possible that we humans have survived so long in a chance filled world, given our evident difficulty with assessing chances correctly?
August 10, November 17, 25, 2021. May 5, 2022.
Copyright, John D. Norton