Four statisticians are playing bridge. One of them says, "I have an ace." The chance that she's holding more than one ace is 5359/14498, which is less than 37 percent.
Later the same player says, "I have the ace of spades." Strangely, the chance that she has more than one ace is now 11686/20825, which is more than 56 percent.
Why does specifying the suit of her ace improve the odds that she's holding more than one ace? Because, though a smaller number of potential hands contain that particular ace, a greater proportion of those hands contain a second ace. It's counterintuitive, but it's true.
That is so deeply counterintuitive that I couldn't explain it to myself. Had to search the web, and found a discussion at Reddit. I feel a little more comfortable now, but counterintuitive things are always particularly hard to digest.
See also these pages in Martin Gardner's Hexaflexagons and other Mathematical Diversions.
I read the comments and like the example of reducing the deck to 3 cards (Ace-Spades, Ace-Hearts, 2-Clubs) and a hand is 2 cards.
ReplyDeleteI think the crux is how the person with the cards reveals the information. If the other person asks "Do you have a spade" versus "Do you have the Ace of Spades" (and the answer is yes), then the probabilities are different. However, because the hand holder can chose whether to say "I have the Ace of Spades" or "I have the Ace of Hearts", it makes a difference to the calculation.
Here are all possible states
#1 #2 prob
--------------
AS AH 0.1666
AH AS 0.1666
AS 2C 0.1666
AH 2C 0.1666
2C AS 0.1666
2C AH 0.1666
Now let's say the person tells you "I have the Ace of Spades". If they have both Aces, then they will say this 50% of the time.
#1 #2 "have AS" orig__ net__
----------------------------------
AS AH 0.50 0.1666 0.833
AH AS 0.50 0.1666 0.833
AS 2 1.00 0.1666 0.166
AH 2 0.00 0.1666 0.000
2 AS 1.00 0.1666 0.166
2 AH 0.00 0.1666 0.000
Given that we heard "I have AS", then the prob they have both Aces is
(0.833 + 0.833)/(0.833+0.833+0.166+0.0166) = 0.333
which is the same as if we heard "I have an Ace". So it isn't a paradox if the person telling you their hand can pick what card they want to mention.
(Sorry about the formatting but blogger doesn't allow table or pre tags)