Start with a bag containing 5 white beans. Randomly draw one at a time employing the following rule:
If the bean is white, color it black and put it back in the bag;
If the bean is black, keep it out.
What is the probability that at some point there will be a single white bean in the bag?
Generalize to start with N beans.
Does the probability converge, and if so, to what value?
(In reply to Analytical solution
by Dan Rosen)
I am afraid it is not that simple. The problem is that not all sequences are equally likely.
Consider n = 2. The correct answer = 1/2, because after one draw the bag has one white bean and one black bean. After two draws the bag either has one white bean or two black beans, each possibility equally likely. There are two different ways of pulling out those last two beans, but that doesn't make any difference to the probability of ending with a white bean.
Or, to show it differently, there are 6 different sequences.
1122 -- prob = 1/4
2211 -- prob = 1/4
1212 -- prob = 1/8
1221 -- prob = 1/8
2121 -- prob = 1/8
2112 -- prob = 1/8
1/3 of them end with 11 or 22, but the correct answer is still 1/2, because some of the sequences are twice as likely as other sequences