It is clear that when you first choose, the probability of choosing the
winning door is 1/3.
If you stick to your first choice the
probability of winning stays at 1/3; nothing
that Monty Hall says changes this probability.
Since there is
a
1/3 probability that your first choice is the prize winning door it follows
that the probability that your first choice is NOT
the prize winning door is 2/3. But if your
first choice is NOT the prize winning door then you are guaranteed
to win by switching doors, That's because Monty Hall always reveals a door without the prize (so, for example, if you chose door 1 and the
prize is behind door 2 Monty Hall would have
to reveal door 3, which has no prize). So,
since you ALWAYS win by switching in the case where your first choice
was not the prize winning door, there is a 2/3 probability of
winning by switching.
It may be easier to appreciate the solution by considering the same
problem with 10 doors instead of just three. In this case there are 9
doors without the prize behind them and one door with the prize. The
contestant picks a door. Monty Hall then has to reveal 8 of
the
other doors without the prize leaving just the contestant's door and
one other door. Unless the contestant chose the one door with the prize
(a 1/10 probability) the only door left unopen by Monty must
be
the one with the prize behind it. So there is 9/10
probability
that the other door will contain the prize, as 9 out of 10 times the
player first picked a door without the prize. The contestant's
probability of winning is therefore 9/10 if he swtiches compared to
1/10 if he does not.
The 'trick' about the three-door game, is that it creates a misleading
impression; the contestant is always presented with 1/3
proportions. There is a 1/3 chance of winning, the host reveals 1/3 of
the mystery, and the player is allowed to switch to the other 1/3
option. All options seem equal — yet they are not.
Another
reason why people overestimate the probability of winning if they stick
is that when Monty Hall reveals that a door that they did not choose
has no prize it tends to confirm in their mind that their first choice
was 'correct'. This is an example of what is commonly referred to as confirmation bias.
It is quite irrational of course because Monty Hall must always reveal
a door without a prize, irrespective of your choice.
A full probabilstic solution using a Bayesian net that you can run
without knowing anything about Bayesian probability can be found by
running the model called "Monty Hall Dilemna", which is in the
Examples/Basic folder of the AgenaRisk
tool.
