Solution
Pólya's Urn
Show the problem again
An urn starts with one red and one blue ball. Repeatedly, a ball is drawn uniformly at random, then returned along with one additional ball of the same color. What is the probability the third draw is red? Express as a fraction.
Worked solution
The answer is 1/2. In fact, every draw in a Pólya urn is marginally red with probability 1/2, by symmetry between the colors (the process is exchangeable). One can verify directly: enumerate the outcomes of the first two draws with their probabilities and sum P(third is red) over them; everything collapses to 1/2. The rich-get-richer dynamics change the joint distribution, not the marginals.
Source: The Polya urn, introduced by Florian Eggenberger and George Polya (1923). Statement written for AxiomIQ.