## Hint for Problem 5

### It is impossible.

Since among X, Y, Z, there is bound to be at least two zeros
or two ones, at least one of the events {X=Y}, {X=Z}, {Y=Z} is
certain to happen. Then since the probability of the union is less
than or equal to the sum of the probabilities,
1 ≤ P(X = Y) + P(X = Z) + P(Y = Z).

### Or is it?

Bell's analysis of the Einstein-Podolsky-Rosen experiment in quantum
mechanics shows that things are not quite so simple. Here is a
probabilist's view of Bell's paradox
- There is an actual physical experiment we may perform that will
produce any of six (dependent) random variables, X, Y, Z, X', Y', Z',
each taking values 0 or 1. This experiment may be replicated
independently any number of times.
- We cannot observe more than two of the variables at the
same time. In fact, we can only observe one primed variable
and one unprimed variable.
- When we observe X and X' together, they always have
the same value, both zero or both one. Probabilistically,
this means P(X = X') = 1. Similarly, P(Y = Y') = 1 and P(Z = Z') = 1.
(Using this, we may reduce the problem to X, Y, Z, by identifying
X with X', Y with Y', and Z with Z'.)
- When X and Y' (= Y) are observed together, we find that
they are the same about 1/4 of the time, and different about
3/4 of the time. So we may presume P(X = Y) = 1/4 approximately.
Similarly, P(X = Z) = 1/4 approximately, and P(Y = Z) = 1/4
approximately.
- But we must have 1 < = P(X = Y) + P(X = Z) + P(Y = Z),
because at least one of the three events {X = Y}, {X = Z},
{Y = Z} must occur. (This is a special case of Bell's inequality.)
So we have
1 ≤ P(X = Y) + P(X = Z) + P(Y = Z) = 3/4 approximately?????!

COMMENT: This paradox does not depend on the assumption that the
experiments are independent and identically distributed.
In fact,
we may imagine a magician assigning his own personal choices of six
zeros and ones and placing them in sealed envelopes labeled
X, Y, Z, X', Y',and Z'. The envelopes are placed on a table in
front of a scientist, who must then select one primed envelope and
one unprimed envelope. Then the magician retains and destroys the
four unselected envelopes, and the scientist opens the two selected
envelopes.

It turns out that if a primed and an unprimed envelope
of the same letter were selected, then the two numbers they contain
will always be the same. However, if a primed and unprimed envelope
of different letters were selected then the numbers they contain
would be the same only about 1/4 of the time.

How can the magician
do it? He seems to be able to guess, with success greater than
possible by chance, whether or not the scientist
is going to choose to observe the same lettered envelopes. He must
have ESP! What does the Amazing Randi say to all this?