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Chapter 6#

Coins#

Suppose a player starts with initial capital \(K\). With each play, with probability \(p\), the player wins, increasing his capital to \(K+1\). If the player looses (with probability \(q=1-p\)), then his capital shrinks to \(K-1\). Play continues until either the player loses all his capital (\(K=0\)), called “ultimate ruin”, or the player wins all of the bank’s capital (\(K=B\)).

Schroeder introduces the “probability of ultimate ruin” \(q_{K}\) which is the probability that the player will eventually lose all of his money. Schroeder describes this by a difference equation

\[\begin{gather*} q_{K} = pq_{K+1} + qq_{K-1}, \qquad 0 < K < B,\\ q_0 = 1, \qquad q_B = 0, \end{gather*}\]

where \(B\) is the total capital in the game (i.e. the capital of the bank). The initial conditions are:

  • \(q_0 = 1\): the player has already lost since he starts with no capital \(K=0\) – certain ruin – and

  • \(q_B = 0\): the player has all the money and has won – no chance of ruin.

Do It! Solve for \(q_K\).

\[\begin{gather*} q_K = \frac{(q/p)^{B} - (q/p)^{K}}{(q/p)^{B} - 1}. \end{gather*}\]

Claude Shannon’s Outguessing Machine#

https://github.com/AnandChowdhary/claude