The seventh ball drawn

Question

A bag holds 4 red balls and 6 white balls, 10 in total. You pull them out one at a time until the bag is empty, laying them in a row in the order drawn.

What is the probability that the seventh ball pulled out is red?

Accepted Answer

It is tempting to grind through the first six draws, but there is a shortcut based on symmetry. Imagine the 10 balls laid out in a random row, position 1 through position 10. Every ordering of the 10 balls is equally likely, so any single position, the seventh included, is just as likely to hold any particular ball as any other position is. In other words, the seventh spot is a fair, random ball out of all 10. So the chance the ball in position 7 is red is simply the fraction of red balls overall: P(	ext{7th is red})=\frac{4}{10}=\frac{2}{5}=0.4. The position number never entered the calcula…

The seventh ball drawn

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