I have three boxes, each with 2 compartments. One has two gold bars, one two silver bars and one has one gold and one silver bar. You choose a box at random and open a compartment at random. If that bar is gold, what is the probability that the other bar in the box is also gold?
Solution
Your first thought is 1 in 2 chance. Since there are only two boxes with a gold bar in it, you reason, I must have picked one of those. Since one has a gold bar and the other has a silver bar on the other side, the probability of having another gold bar is 1/2. Right?
Wrong. It’s actually more complicated than that. To figure out why it’s not a 1 in 2 chance, let’s label the bars like so:
G1
S1
G3
G2
S2
S3
Then let’s enumerate all of the possible draws:
Your first draw
The other side
G1
G2
G2
G1
S1
S2
S2
S1
G3
S3
S3
G3
Next, let’s only focus on draws where the first was gold:
Your first draw
The other side
G1
G2
G2
G1
S1
S2
S2
S1
G3
S3
S3
G3
So, ther’s a 2/3 chance that the other side contains a gold bar given that you drew a gold bar on the first try. 2 out of 3 times you draw another gold bar, because 2 out of 3 times when you picked a gold bar it was either bar #1 or bar #2. 1 out of 3 times you draw a silver bar, because 1 out of 3 times when you picked a gold bar you pick bar #3.
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