Fun with Probability

Prof. Derek Abbott at the University of Adelaide posed a list of questions as "Imponderables". One of the questions, involving probabilities and expectations, caught my eye. I am going to reproduce that question here (edited), along with my "answer" to the question:

If someone comes to you and tells you, "Give me any amount of money you like. I will toss a fair coin. If it is heads, you win and I will double your money; if it is tails, you lose and I will give you only half your money back". You play this game, because if the amount you bet is N, you can win either 2N or N/2, giving an expected return of 1.25N.

Now consider the following related game: Someone presents you with two envelopes and says, "One envelope contains 2N and the other contains N/2. Pick an envelope". Then when you've picked an envelope, this person asks if you want to keep that envelope or swap it for the other one. What strategy should you adopt? Swap or not? Let us complicate it a bit further: Suppose you have the option of looking inside the first envelope. What then?

What is the fundamental difference between the two games described above? Another related question is: does the act of observing the amount in the first envelope change the probabilities in question?

First, some terminology: Let S1 be the strategy of staying with the first envelope, and let S2 be the strategy of swapping the first envelope for the second.

In the version of the game where the player can not see the contents of the first envelope, it is easy to see that a player's expected earning is 1.25N with either strategy S1 or S2. Let us focus on the version of the game where the player is allowed to see the amount in the first envelope.

Consider that you are the player, but you have short-term amnesia, rendering you unable to remember the value of N from the previous time you played the game.

Suppose you play the game 20 times, following the strategy S1. If the first envelope is chosen at random from the two envelopes, you can expect the first envelope to contain N/2 in about 10 games, and 2N in the other 10 games. In fact, this is true a priori: It is true even before you see the amount in the first envelope. It follows that your average earning per game is 1.25N.

Similarly, suppose you play the game 20 times, following the strategy S2. Observe that if the first envelope is chosen at random like before, you can expect that the second envelope contains N/2 in 10 games and 2N in the other 10 games. Once again, this is a priori true. Seeing the amount in the first envelope does not change the fact. Clearly, your average earning per game is 1.25N again.

So, both strategies are equally beneficial, in both versions of the game. To be formal about it, let E[S] be the expected return for strategy S. Let Y and Z be random variables representing the amounts inside the first and the second envelopes, respectively.

Clearly, Pr(Y = N/2) = Pr(Y = 2N) = Pr(Z = N/2) = Pr(Z = 2N) = 1/2.

Let F be the event that the player looks inside the first envelope. The crucial observation is that the event Y = N/2 is independent of F. So are the events Y = 2N, Z = N/2 and Z = 2N.

Therefore, in both versions of the game,
E[S1] = E[Y] = 1.25N, and
E[S2] = E[Z] = 1.25N.