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The necktie paradox is a puzzle and
The necktie paradox is a puzzle and paradox
A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-contradictory or a logically u ...
with a subjective interpretation of probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
describing a paradoxical bet advantageous to both involved parties. The two-envelope paradox
The two envelopes problem, also known as the exchange paradox, is a paradox in probability theory. It is of special interest in decision theory, and for the Bayesian interpretation of probability theory. It is a variant of an older problem known ...
is a variation of the necktie paradox.
Statement of paradox
Two persons, each given a necktie, start arguing over who has the cheaper one. The person with the more expensive necktie must give it to the other person. The first person reasons as follows: winning and losing are equally likely. If I lose, then I will lose the value of my necktie. But if I win, then I will win more than the value of my necktie. Therefore, the wager is to my advantage. The second person can consider the wager in exactly the same way; thus, paradoxically, it seems both persons have the advantage in the bet.Resolution using fluid intelligence
The paradox can be resolved by giving more careful consideration to what is lost in one scenario ("the value of my necktie") and what is won in the other ("more than the value of my necktie"). If one assumes for simplicity that the only possible necktie prices are $20 and $40, and that a man has equal chances of having a $20 or $40 necktie, then four outcomes (all equally likely) are possible: The first man has a 50% chance of a neutral outcome, a 25% chance of gaining a necktie worth $40, and a 25% chance of losing a necktie worth $40. Turning to the losing and winning scenarios: if the man loses $40, then it is true that he has lost the value of his necktie; and if he gains $40, then it is true that he has gained more than the value of his necktie. The win and the loss are equally likely, but what we call "the value of his necktie" in the losing scenario is ''the same amount'' as what we call "more than the value of his necktie" in the winning scenario. Accordingly, neither man has the advantage in the wager. This paradox is a rephrasing of the simplest case of thetwo envelopes problem
The two envelopes problem, also known as the exchange paradox, is a paradox in probability theory. It is of special interest in decision theory, and for the Bayesian interpretation of probability theory. It is a variant of an older problem known ...
, and the explanation of the resolution is essentially the same.
See also
*Bayesian probability
Bayesian probability is an interpretation of the concept of probability, in which, instead of frequency or propensity of some phenomenon, probability is interpreted as reasonable expectation representing a state of knowledge or as quantification ...
* Bertrand paradox
* Decision theory
Decision theory (or the theory of choice; not to be confused with choice theory) is a branch of applied probability theory concerned with the theory of making decisions based on assigning probabilities to various factors and assigning numerical ...
* Monty Hall problem
The Monty Hall problem is a brain teaser, in the form of a probability puzzle, loosely based on the American television game show ''Let's Make a Deal'' and named after its original host, Monty Hall. The problem was originally posed (and solved) ...
* Two envelopes problem
The two envelopes problem, also known as the exchange paradox, is a paradox in probability theory. It is of special interest in decision theory, and for the Bayesian interpretation of probability theory. It is a variant of an older problem known ...
* Newcomb's paradox
In philosophy and mathematics, Newcomb's paradox, also known as Newcomb's problem, is a thought experiment involving a game between two players, one of whom is able to predict the future.
Newcomb's paradox was created by William Newcomb of the ...
* St. Petersburg paradox
The St. Petersburg paradox or St. Petersburg lottery is a paradox involving the game of flipping a coin where the expected payoff of the theoretical lottery game approaches infinity but nevertheless seems to be worth only a very small amount to t ...
References
* {{cite book , first=Maurice , last=Kraitchik , authorlink=Maurice Kraitchik , title="Mathematical Recreations" , publisher=George Allen & Unwin , location=London , year=1943 Probability theory paradoxes Decision-making paradoxes