Everyone will agree with this one, casinos are the most depressing places in the world. Well, consider Casino Royale, where you can find a game galled Birawa. Its principle is quite simple. You have a certain capital (an integer number of tokens, say) and the dealer throws a coin who eithers falls on heads, with odds p₁=1/2+ε, or on tails, with odds q₁=1-p₁=1/2-ε; with ε=0.005. If the outcome is tails, you win one token, in the opposite case you lose one.

Q1: show that Birawa is a losing game, i.e. that playing it an infinite number of times results in the loss of the initial capital. Bonus question: explain the game's name.

In the casino stands also this game called Murk, where the rules are following.
If the capital at your disposal is a multiple of three, the dealer throws a coin with odds q₂=1/10-ε to fall on tails (resulting in winning one token). Otherwise he throws a coin with odds q₃=3/4-ε to fall on tails. We still have ε=0.005 and the fact that a "heads" outcome leads to surrender one token.

Q2 : show that Murk is also a losing game, i.e. that playing it an infinite number of times also results in the inescapable loss of the initial capital. Markov Chains Theory may be invoked.

Therefore, like in every casino, each game favours (slightly) the House – the so-called house edge. Which means, from the collective point of view of all the players, more losses than earnings.

Q3 : Show that nevertheless, it might be interesting to go to Casino Royale.