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Tricky probability puzzle

Posted By: Jason Lee
Date: Saturday, 28 October 2023, at 5:01 p.m.

In Response To: Tricky probability puzzle (Timothy Chow)

I believe every card is equally likely to be dealt. To test this, I took a simplified deck of 12 cards that only has aces, kings, and queens (four each). I wrote a Python program to check all 12! permutations of the the cards -- not a simulation (or rollout) -- this is as exact calculation. Results: all cards were equally likely to turn up after the first ace.

There are some subtleties in questions like this. There is a related one where you deal me two cards at random. I look at my cards and say to you, "One of my cards is an ace. What is the probability my other card is an ace?" You could work it out, it's not a hard problem in conditional probability.

Here's where things get weird. I look at my cards and say to you, "One of my cards is the ace of spades. What is the probability my other card is an ace?" I still don't get this -- the answer is different. When I say I don't get it, I can compute both of these -- I just don't have any intuition why it should be different -- the card I look at has to be a particular suit, why does KNOWING the suit change the probability? My answer is that the human brain is ill suited to have intuition about conditional probability, which is precisely why people don't "get" the Monty Hall Paradox.

JLee

P.S. My program to compute the probabilities for the full deck of 52 cards is still running. I will let you know when it finishes.

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