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, 2006

Combinatorics Difficulty 3.0 AMC 10/12 Prove it Canada

A bag contains some blue and some green hats. On each turn, Julia removes one hat without looking, with each hat in the bag being equally likely to be chosen. If it is green, she adds a blue hat into the bag from her supply of extra hats, and if it is blue, she adds a green hat to the bag. The bag initially contains $4$ blue hats and $2$ green hats. What is the probability that the bag again contains $4$ blue hats and $2$ green hats after two turns?

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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project. Solutions are the publisher's, linked not copied.