Replacing urn balls with blue until all blue

Question

An urn holds 2 red balls and 1 blue ball. Each step you draw a ball uniformly at random and put back a blue ball in its place, whatever colour you drew. What is the expected number of draws until every ball in the urn is blue?

Short answer

$9/2$ . With 2 reds the next draw is red with probability $\tfrac23$ (mean wait $\tfrac32$ draws); with 1 red it is $\tfrac13$ (mean wait 3). The two geometric waits add to $\tfrac32+3=\tfrac92$ .

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Lessons behind this question

Expectation, Mean, and Variance
Conditional Expectation and Variance Revisited
Continuous Random Variables and Probability Density Functions

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