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:
- Select one box uniformly at random. Pull one ball from that box. Or,
- 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?