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 Four balls (Posted on 2017-05-19)
There are four balls in a hat: a blue one, a white one, and two red ones. Now I draw simultaneously two balls, look at them, and announce that at least one of them is red.

What is the chance that the other is red as well?

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 re: Who's right? | Comment 9 of 22 |

Is it so simple?

One method includes numbering the red balls 1 and 2. I'm not sure if that is right, if they are identical. But assume it is, and they are red 1 and red 2:

permute 2 from {1,2,3,4} {1, 2} | {1, 3} | {1, 4} | {2, 1} | {2, 3} | {2, 4} | {3, 1} | {3, 2} | {3, 4} | {4, 1} | {4, 2} | {4, 3} (total: 12)

Order doesn't matter, so we obtain:{1, 2} {1, 3}{1, 4}{2, 3}{2, 4}  {3, 4} (in some order). This results in the 1/5 answer.

But since the reds are identical:

permute 2 from {1,1,3,4} {1, 1} | {1, 3} | {1, 4} | {3, 1} | {3, 4} | {4, 1} | {4, 3} (total: 7)

Order doesn't matter, so we obtain: {1,1,3,4} {1, 1} {1, 3} {1, 4} {4, 3} (in some order). This gives the 1/3 answer.

But if we consider 'red and non-red', it is different again:

permute 2 from {1,1,2,2} {1, 1} | {1, 2} | {2, 1} | {2, 2} (total: 4)

Order doesn't matter, so we obtain: {1,1,2,2} {1, 1}{1, 2}  {2, 2}. This gives an answer of 1/2.

All that being said, I did experiment with 2 blue balls (B) and one yellow one (Y) and the results of 100 draws were 63BY and 37BB. By parity of reasoning, Nature does seem to distinguish between ball Blue 1 and Blue 2, so the 1/5 answer would seem to be preferable.

Edited on May 20, 2017, 3:00 am
 Posted by broll on 2017-05-20 02:56:51

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