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?
The first draw, we get a white bean. We color it black and put it back in the bag. There are now 4 white beans.
The second draw, we either:
1. draw the black bean, discard it, and then draw a white bean, color it black and put it back. If this happens (2 draws), there are now 3 white beans.
2. draw a white bean, color it black, and return it to the bag. There are now 3 white beans in this case too.
By extension this same scenario applies every time. We either draw black beans until they are gone, only to decrease the number of white beans by one on the next draw, or we directly decrease the number of white beans by one on the first draw.
After this has happened two more times past what has been described, there is one white bean in the bag with certainty one.
Edited on July 29, 2015, 2:00 pm