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\title{Probability - you think you understand it?}
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\section{Prefatory Note}
This note contains a collection of paradoxes on probability. Each
paradox is ranked on a scale from 1 (least difficult) to 5 (most
difficult).  A short explanation of the source of the confusion is
presented In addition, where possible the source of the problem is
also referenced.

\section{Paradoxes}


{\bf 1. The envelope-swap paradox [3] : }  I have two envelopes, the one
containing \$$x$ and the other containing \$$2x$, where $x$ is any real
positive number. I shuffle the two envelopes and give one envelope to
you and one to your friend at random. Neither of you is allowed to
open your respective envelopes at this point. Now I ask you - do you
want to swap your envelope with your friend. You say yes because if
your envelope contained \$$y$ then with probability $\frac{1}{2}$ your
friend received \$$2y$ and with probability $\frac{1}{2}$ she received
\$$\frac{y}{2}$ and so your expected payoff is \$$\frac{1}{2}\times 2y
+ \frac{1}{2}\times \frac{y}{2} = 1.25y$. Your friend reasons the same
way and wishes to swap envelopes too. But there has to be something
wrong here, this can't be a win-win situation for both of you. Further
if you keep swapping it seems you are increasing your expected payoff
each time. Can this be possible? Is this what is happening on Wall
Street for example, where nobody is creating any real wealth but by
just swapping stuff everyone becomes better off?

{\it Explanation : } If you received \$$y$ then either $y=x$ with
probability a $\frac{1}{2}$ or $y=2x$ with probability a
$\frac{1}{2}$. So the expected gain is \$$x$ with probability a
$\frac{1}{2}$ or \$$-x$ with probability a $\frac{1}{2}$ and hence the
expected payoff is \$$0$ the way one expects it.

{\bf 2. The information paradox [4] : } I turn two cards facedown in front
of you. Each card has a number written on it. You win if you pick the
card with the larger number. So a randomized strategy that ensures a
win at least (and in fact exactly) half the time is one where you pick
a card at random. Suppose I now tell you that you can turn over one of
the cards, look at the number and then decide to make a switch if you
want to, is there any way you can improve your chances of winning to
more than a half. In other words is there any value to the extra
information you now have namely the value of the number one of the
cards.

{\it Explanation : } Believe it or not, there is such a strategy. Pick
any continuous everywhere non-zero distribution, pick a number
according to this distribution, then turn over a card at random,
compare the two numbers, if the number on the card is less than the
number picked from the distribution then switch.

{\bf 3. The expectation paradox [4] : } There is some fixed
distribution over the reals. I pick a number from this distribution.
Now you have to keep picking numbers (independently) from this same
distribution until you pick a number greater than or equal to mine.
You have to pay me one dollar for each number you pick. How many
dollars must I pay you for the privilege of playing this game.

{\it Explanation : } Again believe it or not, it is not \$2 or some
such ammount. The answer is infinity, and this is true for all
distributions.

{\bf 4. The Monty Hall paradox [2] : }

{\bf 5. Raymond's paradox - chords in a circle[3] : }

{\bf 6. The red/black cards problem [3] : }

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