Waiting for the first red ball
An urn contains red balls and blue balls. Balls are drawn one at a time, without replacement, until the first red ball appears.
Find the expected number of draws, the red ball included.
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
The number of draws is plus the number of blue balls drawn before the first red one. Use one indicator per blue ball.
For a given blue ball, look only at the relative order of that ball and the three red balls. In how many of the equally likely positions does it come first?
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s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. Count the blue balls that come before every red ball. For a single blue ball this is a question about the order of just four balls, which is answered by symmetry.
Step 1: indicators. Number the blue balls to and let if blue ball is drawn before all three red balls. The number of draws is
Step 2: one blue ball. The draw order is a uniformly random ordering of the balls. Restricted to blue ball and the three red balls, every one of the balls is equally likely to come first. So .
Step 3: add.
In general. With red and blue balls, the expected position of the first red ball is . The red balls split the other balls into gaps of equal expected size.