You drew a gold coin, is the chest's other coin gold?

Question

There are three chests, each holding two coins. One chest has two gold coins, one has two silver coins, and one has one gold and one silver coin. You pick a chest at random, reach in, and draw one coin. It is gold.

What is the probability the other coin in that chest is also gold?

Accepted Answer

The naive answer 1/2 (it is either the gold-gold chest or the gold-silver chest) overlooks how many gold coins each chest offers you to draw. Count individual gold coins instead. There are three gold coins across the chests: two in the gold-gold chest and one in the gold-silver chest. Drawing a gold coin means you are equally likely to be holding any of these three. - Two of them come from the gold-gold chest, whose other coin is gold. - One comes from the gold-silver chest, whose other coin is silver. P(	ext{other gold} \mid 	ext{drew gold}) = \frac{2}{3} \approx 0.667. The all-gold chest …

You drew a gold coin, is the chest's other coin gold?

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