A and B are both losing games.

A is simple, B is complicated.

In simple game A, you win $1 with probability p (a bit less than 0.5, say 0.495) and lose $1 with probability 1-p (0.505).

Game B is a combination of two games B_{1} and B_{2}.

In B_{1}, you win $1 with probability p_{1} (a bit less than 0.1, say 0.095). You lose $1 with probability 1−p_{1} (0.905).

In B_{2}, you win $1 with probability p_{2} (a bit less than 0.75, say 0.745). You lose $1 with probability 1−p_{2} (0.255).

The instructions for B are: play B_{2} if your stash is a multiple of 3, B_{1} otherwise.

A is a losing game, B is a losing game, but the combination ABBABABBABABBABABBAB… is winning.

Click here for simulation.

And see the map of the “probability space” below.

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