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 p_{1}=1/2+ε, or on tails, with odds q_{1}=1-p_{1}=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 q_{2}=1/10-ε to fall on tails (resulting in winning one token). Otherwise he throws a coin with odds q_{3}=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.*

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 q

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.

Win Win Win! [Prison Break 2x21]

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