Explaining the Second Ace Paradox: Understanding the Math Behind It

[gsingh2011]

Could someone please explain the math behind the Second Ace Paradox? Here's the problem: http://www.futilitycloset.com/2009/02/06/the-paradox-of-the-second-ace/

I want to know how the poster got those probabilities... I can't figure it out on my own.

 

[Hurkyl]

There's an important part of the problem missing.

But anyways, what the poster considered was:

(number of hands with the ace of spades and two or more aces) / (number of hands with the ace of spades)​

and

(number of hands with two or more aces) / (number of hands with one or more aces)​

There is a pseudo-paradox in the fact that the statement of the problem strongly suggests that something else entirely is meant: that the other player simply picked one of the aces he has and named it.

It's much clearer if you think of you interrogating the player with the series of questions:

• Do you have an ace?
• Do you have the ace of spades?

and getting two "yes" answers, versus

• Do you have an ace?
• If so, name an ace in your hand

and getting "yes" and "spade" answers.

In the first series of queries, the answer to the second question clearly gives information as to whether or not the player has two aces. In the second series of queries, the answer to the second question clearly gives no information.

 

[gsingh2011]

Thanks for replying.

I don't see why those two cases you mentioned are different; they seem the same to me. Can you explain?

 

[D H]

Hurkyl's first set of questions limits the statistical universe to the set of randomly dealt bridge hands that contain the ace of space. Amongst this statistical universe, the probability of such a hand containing at least one other ace is 11686/20825.

In Hurkyl's second set of questions, the person who said "yes" to "do you have an ace" could have answered (for example) "the ace of hearts" to the second question. The second questions adds no new information. It does nothing to cut down the size or distribution of the statistical universe. The probability of a second ace given that the answer to "do you have an ace" is "yes" is 5359/14498. Since the second question here adds no new information, the probability of a second ace remains at 5359/14498.

 

[gsingh2011]

Ok, I've figured out the math behind the original question, but the difference between the first and second set of questions Hurkyl posted has intrigued me and confused me. Why does the second question "add no new information"? With the first set of questions, you know that the person has at least one ace, and one of those aces is the ace of spades. Isn't that the case in the second set of questions? You know the person has an ace, and the name of one of those aces is the ace of spades. So why doesn't that decrease the statistical universe?

 

[D H]

Suppose you have a bridge that contains an ace. Telling me that you have the ace of spades, the ace of hearts, the ace of diamonds, or the ace of clubs adds zero information. I already knew that you had at least one of these four cards when you answered "yes" to "do you have an ace".

 

[D H]

Another take at it: The article would have been right if it simply talked about the probability of a second ace given (a) a randomly-dealt bridge hand that contains an ace versus (b) a randomly-dealt bridge hand that contains the ace of spades.

By making this dry problem into a story the author of the article has introduced a flaw. As written, I would say the article is incorrect.