All red marbles out before any blue

Question

A jar holds 3 red marbles and 3 blue marbles, 6 in total. You draw them one at a time until the jar is empty. What is the probability that all three red marbles come out before any blue marble?

Accepted Answer

Consider the whole color sequence. Picture the 6 marbles in a random row of 6 positions; only which 3 positions are red matters. Every choice of "which 3 positions are red" is equally likely, and there are \binom{6}{3}=20 of them. "All reds before any blue" is exactly one pattern: reds take the first three positions. So P=\frac{1}{\binom{6}{3}}=\frac{1}{20}=0.05. Draw-by-draw check: \frac{3}{6}	imes\frac{2}{5}	imes\frac{1}{4}=\frac{6}{120}=\frac{1}{20}. Takeaway. Even with an even split of colors, a clean sweep of one color first is a 1-in-20 event here: one favorable pattern out of \binom{…

All red marbles out before any blue

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