Category Archives: Bayesian

The Exchange Paradox

Photograph by Jake Slagle @ flickr

 

“A swami puts m dollars in one envelope and 2m dollars in another. You and your opponent each get one of the envelopes (at random). You open your envelope and find x dollars, and then the swami asks you if you want to trade envelopes. You reason that if you switch, you will get either x/2 or 2x dollars, each with probability 1/2. This makes the expected value of a switch equal to (1/2)(x/2) + (1/2)(2x)=5x/4, which is greater than the x dollars that you hold in your hand. So you offer to trade.

 

The paradox is that your opponent has done the same calculation. How can the trade be advantageous for both of you?”

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The Maths of Paul the “Psychic” Octopus

David Spiegelhalter writes:

England’s performance in the World-Cup last summer was thankfully overshadowed by the attention given to Paul the Octopus, who was reported as making an unbroken series of correct predictions of match winners. Here we present a mathematical analysis of Paul’s performance in an attempt to answer the question that (briefly) gripped the world: was Paul psychic?    more

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The Monty Hall problem

Consider the following problem
Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?

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(Formulation by Marilyn vos Savant at Whitaker, Craig F. (1990).)
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So what is the answer? Let’s see a simple solution. The following table shows the three possible arrangements of two goats and one car behind three closed doors and the result of switching or staying after initially picking Door 1

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Door 1 Door 2 Door 3 Outcome if switching Outcome if staying
Car Goat Goat Goat Car
Goat Car Goat Car Goat
Goat Goat Car Car Goat

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Which means that in 2 out of 3 cases we would win the car by switching doors. Hence it would make perfect sense to always switch. To understand why this is happening do take into consideration that the event “staying” is composed by only one element while its complement by two (note that all elements are equiprobable).

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Frequency of winning when staying/switching doors in a sequence of "Monty-Hall" games

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Bayesians vs. Frequentists


Lecture by Michael I. Jordan


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