Let P (A|C) be the probability that the prize is
in box A given that the host opens box C. Using
Bayes Theorem,
we get,
If the grand prize is in box A, the probability
that it is in box A given that the host opens box C is
1/3. To calculate the probability that the prize is in
box B, we use Bayes theorem again to give:
P(B) = 1/3 and P(C) = 1/2. If the grand prize is in
box C, then the host can only open box B, giving
us P(C|B) = 1. Substituting these values we obtain:
Thus, swapping gives us a 2/3 chance of
selecting the grand prize, doubling our chances.
Still unconvinced? Let’s look at an example where
there are 100 boxes. The probability of picking
a grand prize is 0.01 (1%) and the probability of
picking a goat is 0.99 (99%). Thus, the only time
swapping is a bad idea is if you had already
picked the winning box, provided that the host
shows you 98 other boxes with goats.
It was shown that the simple pigeon adapted
to the Monty Hall Problem surprisingly well [3].
In an animal experiment, pigeons were shown
three computer-controlled lit keys, one of which
contained food. On the first pigeon peck, the
computer would switch off all three keys but
would then switch two back on shortly afterward,
one of which was the pigeon’s original choice.
The pigeon was then awarded food if it selected
the correct key in the remaining lit keys. At the
beginning, only one third of the pigeons switched
to a different key. However, after one month, all six
pigeons involved in the study switched their choice
consistently to obtain the highest chance of
selecting food, indicating that switching became
a learned behaviour from reinforcement.
In a similar set up, students were also given
three lit keys and instructed to obtain the highest
points. Again, over one month, they were to guess
the right keys, limited to 200 tries. Their results
unfortunately did not measure up to the pigeons.
Equally likely to switch or stick to their original
choices in the beginning, there was little sign of
improvement toward the end of the month.
While this experiment does not indicate that
pigeons are smarter than humans, it does show us
that we tend to do a lot of overthinking, perhaps
leading to false reasoning. Interestingly enough, it
was the youngest students who fared best in this
experiment, and the older students were more
likely to overthink. Not too surprising, considering it
was claimed that Nobel physicists routinely gave
the wrong answer to the Monty Hall Problem!
