Ball in Boxes

Suppose you have three boxes, \(Box_1,Box_2,Box_3\), such that \(Box_i\) contains \(i\) white balls and one black ball.

You will to select one ball from the boxes. Here are two schemes you could use for selection:

  1. Select one box uniformly at random. Pull one ball from that box. Or,
  2. Dump all the balls into one box. Mix them up. Pull out one ball.

Are these two schemes probabilistically equivalent?

Suppose instead of selecting a box uniformly at random, you select \(Box_i\) with probability \(p_i\). Find a list of values for \(p_1, p_2,\) and \(p_3\) that would make this new scheme probabilistically equivalent to scheme 2?

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