A Polya urn starts with 2 red and 3 blue balls. A ball is drawn…
Question
A Polya urn starts with 2 red and 3 blue balls. A ball is drawn uniformly at random; its colour is noted, and then it is returned to the urn together with one additional ball of the SAME colour. This procedure is repeated for a total of 4 draws. Let the probability that exactly 2 of the 4 drawn balls are red be m/n in lowest terms. Find m+n.
Step-by-step solution
Each ordered colour sequence of 4 draws with exactly 2 reds has its probability computed by multiplying sequential conditional probabilities, where after drawing a colour its count rises by 1 and the total rises by 1.
A key property of the Polya urn: for a fixed number of reds, every order of the same multiset of colours gives the SAME probability (the numerator factors of red counts {2,3} and blue counts {3,4} appear in some order, and the denominators are always 5,6,7,8).
So compute one order and multiply by the number of arrangements.
Final answer44
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