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Conditioning and Polya’s urn
An urn contains 1 black and 2 white balls. One ball is drawn at random and its color noted. The ball is replaced in the urn, together with an additional ball of its color. There are now four balls in the urn. Again, one ball is drawn at random from the urn, then replaced along with an additional ball of its color. The process continues in this way.
- Let \(B_n\) be the number of black balls in the urn just before the \(n\)th ball is drawn. (Thus \(B_1= 1\).) For \(n \geq 1\), find \(\mathbf{E} (B_{n+1} | B_{n}) \).
- For \(n \geq 1\), find \(\mathbf{E} (B_{n}) \). [Hint: Use induction based on the previous answer and the fact that \(\mathbf{E}(B_1) =1\)]
- For \(n \geq 1\), what is the expected proportion of black balls in the urn just before the \(n\)th ball is drawn ?
[From pitman p 408, #6]
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.
- Calculate \(\mathbf{P}(\text{Black wins})\) and \(\mathbf{P}(\text{White wins})\) in terms of \(p=b/(b+w)\).
- What value of \(p\) makes the game fair (equal chances of wining) ?
- Is the game ever fair ?
- 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]
Polya’s urn
An urn contains \(4\) white balls and \(6\) black balls. A ball is chosen at random, and its color is noted. The ball is then replaced, along with \(3\) more balls of the same color. Then another ball is drawn at random from the urn.
- Find the chance that the second ball drawn is white.
- Given the second ball drawn is white, what is the probability that the first ball drawn is black ?
- Suppose the original contents of the urn are \(w\) white and \(b\) black balls. Also after drawing a ball we replace with \(d\) balls of the same color. What is the probability that the second ball drawn is white (it should be \(\frac{w}{w+b}\) )?
[Pitman page 53. Problem 2]
