Ludwik reprises the classic "Monty Hall" problem. It's
interesting to examine a few unspoken assumptions in "the
solution."
The usual assumption is that Monty *always* shows you a goat
(because Monty *always* knows where there is a goat) and *always*
offers you a chance to switch. In this case, you win 1/3 of the
time if you never switch and 2/3 of the time if you always switch.
The reason is that Monty's actions provide you with *extremely*
valuable information: After all, *every* time you initially pick
a goat (i.e., 2/3 of the time), Monty then tells you precisely
where the car is.
But what if Monty *only* shows you a goat and offers you a chance
to switch when you have chosen the car? Then the best strategy
is *never* to switch and you end up winning 1/3 of the time--i.e.,
*every* time you are given the chance to switch.
And what if Monty *only* shows you a goat and offers you a chance
to switch when you have chosen a goat? Then the best strategy is
*always* to switch when offered the chance and the result is that
you win 100% of the time.
And what if Monty *always* opens a door and offers you the chance
to switch but does so without knowing what is behind the doors?
Then, 1/3 of the time he shows you a car and you win a goat no
matter what you do. The other 2/3 of the time you win a car with
50% probability whether you do or don't switch. In other words,
you win 1/3 of the time. This is because Monty gives you no
useful information.
In general, Monty isn't likely to follow *any* predictable pattern
of behavior, and the odds of winning the car will probably remain
about 1/3 no matter what you do as well.