First red ball on the third draw

Question

A bag holds 2 red balls and 6 blue balls, 8 in total. You draw balls one at a time, without replacement. What is the probability that the first red ball appears on exactly the third draw?

Accepted Answer

"First red on the third draw" means the first two are blue and the third is red. Chain the draws, shrinking the bag each time. Draws 1 and 2 both blue. Blue starts at 6 of 8, then 5 of 7: \frac{6}{8}	imes\frac{5}{7}. Draw 3 is red. After 2 blues leave, both reds remain out of the 6 balls left: \frac{2}{6}. Multiply: \frac{6}{8}	imes\frac{5}{7}	imes\frac{2}{6}=\frac{60}{336}=\frac{5}{28}\approx 0.179. Takeaway. Force the failures first, then the success. With only two reds available, the third-draw red still counts on both being present after two blue removals.

First red ball on the third draw

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