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Games with Black and White Balls

Consider the following gambling game for two players, Black and White. Black puts \(b\) black balls and White puts \(w\) white balls in a box. Black and White take turns at drawing randomly from the box, with replacement between draws until either Black wins by drawing a black ball or White wins by drawing a white ball. Suppose Black gets to draw first.

  1. Calculate \(\mathbf{P}(\text{Black wins})\) and \(\mathbf{P}(\text{White wins})\) in terms of \(p=b/(b+w)\).
  2. What value of \(p\) makes the game fair (equal chances of wining) ?
  3. Is the game ever fair ?
  4. What is the least total number of balls in the game, \((b+w)\), such that neither player has more that that \(51\%\) chance of winning ?

 

[Pitman P219, #13]

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