Probability of an expected outcome

Question

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?

Answer 1

There is little typo in your question. There should be $$ 800 \cdot \frac{1}{2} + 3200 \cdot \frac{1}{2} = 2000. $$ What your professor said is that the expected value of your winning is 2000 when you change the doors. In some way it means that when you play this game many times then your average winning is going to be closer and closer to 2000 (assuming that you always chose the door with 1600 first).

Formally, the Law of large numbers plays role here. If you take $X_i$ to be your winning in $i$'th game and $S_n$ be the sum of your winnings in $n$ games then $$\frac{S_n}{n} \to 2000.$$

Hence if you change the door then your average win will be greater then 1600. Here it is important that you either win 1600 more or 800 less then in your first choice with equal probability.

Answer 2

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.