I'm in a class titled "Puzzle Based Learning" and we were given this problem:
There is a new game show and you are
the participant. There are two doors,
each has a suitcase with gold coins
behind it. You know that these
suitcases contain amounts from the
set: 25, 50, 100, 200, 400, 800,
1,600, 3,200, and 6,400. You also know
that one suitcase has twice as much
coins as the other. You select the
door and open the suitcase. You find
1,600 gold coins. Then the game show
host offers you the opportunity to
switch doors. Do you do it? Justify
your answer.
In class, the professor tells us:
The two possibilities of the suitcase
are 800 and 3200. The expected
outcome of choosing one suitcase would
be 800*.5 + 3200*.5 = 2000. Since
2000>1600, you should choose again
because the expected outcome is
greater than 1,600.
I was hoping someone could explain this a little more clearly. In my mind, it seems there is a 50/50 chance of getting 800 and 3200 and the "expected outcome" is meaningless because we shouldn't care about the payoff of choosing again. You have a 50% chance of losing.
Am I missing something? Is this a trick question? Is my professor pulling my leg?
Best Answer
There is something missing, namely any statement that the 16 ways of putting the suitcases behind the doors are equally likely. If they are equally likely then you should always switch after opening a door unless you see 6400, for the reason given.
But suppose that you know each pair of suitcases $(n,2n)$ is three times as likely as the next pair up $(2n, 4n)$. In your particular example the choice would be between $1600$ and an expected $800\times \frac{3}{4} + 3200\times \frac{1}{4} = 1400$ so you would stay with what you see first; similarly for any amount you would keep what you see, unless you see 25 in which case you would switch doors.
If you simply don't know - it is a new game show - then you have to judge the psychology of the game designers. They may think more big winners will attract more viewers and so pay off in advertising; or they may just want to drive down prize costs. Your call.