Pólya's urn is a classic example of reinforced randomness: whenever a particular color comes up, it becomes slightly more likely to come up again. The question is whether that cumulative bias distorts the final distribution or whether, counterintuitively, it ends up not mattering.
The urn bet
Riddle statement
A box starts with 1 red ball and 1 blue ball.
On each turn:
1. A ball is drawn at random;
2. It is returned to the box;
3. a new ball of the same color is added.
After \(n\) turns, there will be \(n+2\) balls in total.
Maria bets that, after \(n\) turns, all possible values for the number of red balls are equally likely. Luis says that the middle values should come up more often.
Who wins the bet? And, more precisely, what is the probability of ending up with exactly \(k\) red balls?