The original games in brief

We introduce two games, A and B. Game A can be described like tossing a coin:

	Game A
	Probability of winning	Probability of losing
	1/2 - e	1/2 + e

Here winning means that the player receives one euro and loosing means that the player loses one euro. Game B is a little more complex and can be described by the following statement: if the present capital is a multiple of 3, then the chance of winning is p₃=1/10-e; if not, then the chance of winning is p₂=3/4-e:

Game B
Is the capital a multiple of 3?
NO			YES
Coin 2	Probability of winning	Probability of losing	Coin 3	Probability of winning	Probability of losing
	3/4 - e	1/4 + e		1/10 - e	9/10 + e

The rules can be summarized by the following diagram:

where p₁=1/2-e, p₂=3/4-e, and p₃=1/10-e. We have used the green and red colors for the coins to indicate that coins 1 and 3 are "bad" coins for the player, whereas coin 2 is "good" for the player.

If we run some simulations playing these games, we find that when played individually the two games lose when e > 0. But consider the scenario if we start switching between the two losing games, play two games of A, two games of B, two of A, and so on. The result, which is quite counter intuitive, is that we start winning. That is, we can play the two losing games A and B in such a way as to make a winning outcome. The result of playing the games is shown in this figure:

Figure 3. The main plot shows the effect of switching between games A and B. The simulation was performed by playing game A a times, then game B b times, game A a times, and so on with e = 0.005 and averaged over 50,000 runs. The notation v =[a,b] is used for each combination