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Shlomi Fish  committed 3f37290

Started writing the analysis.

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File project-euler/121/121-planning.txt

+Problem:
+--------
+
+{{{
+A bag contains one red disc and one blue disc. In a game of chance a player
+takes a disc at random and its colour is noted. After each turn the disc is
+returned to the bag, an extra red disc is added, and another disc is taken at
+random.
+
+The player pays £1 to play and wins if they have taken more blue discs than red
+discs at the end of the game.
+
+If the game is played for four turns, the probability of a player winning is
+exactly 11/120, and so the maximum prize fund the banker should allocate for
+winning in this game would be £10 before they would expect to incur a loss.
+Note that any payout will be a whole number of pounds and also includes the
+original £1 paid to play the game, so in the example given the player actually
+wins £9.
+
+Find the maximum prize fund that should be allocated to a single game in which
+fifteen turns are played.
+}}}
+
+There are two possible states:
+
+S1) 1 B + 1 R .
+
+S2) 2 R.
+
+The state machine is:
+
+S1 -> S1 - 2/3
+
+S1 -> S2 - 1/3
+
+S2 -> S1 - 0
+
+S2 -> S2 - 1/1
+
+The output is:
+
+Out[S1] = B -> 1/2
+Out[S1] = R -> 1/2
+Out[S2] = B -> 0
+Out[S3] = R -> 1/1
+