Near the end of *Lady Luck*, Warren Weaver poses what he calls "the problem of the second ace," which he takes from Martin Gardner's *Scientific American Book of Mathematical Puzzles and Diversions*. You are playing bridge, and one of the other players, having inspected her cards, announces, "I have an ace." What is the probability that she has another ace?

This must be the ratio of the probability that she has more than one ace to the probability that she has at least one ace. Let us just horse it and calculate the probabilities of having any possible number of aces.

[Horse it.]

Probability of having no ace [P(0)] turns out to be about 0.3038; P(1) is about 0.4388; P(2) about 0.2135; P(3) about 0.0412; and P(4) about 0.0026. We said the answer to the original problem is given by [P(2) + P(3) + P(4)]/[1 - P(0)], which, plugging in our values and pushing the right buttons on a calculator, we learn is just about exactly 0.37-- the answer, says Weaver, supplied by Gardner.

There is a twist to the problem that is more interesting. A few hands later, the same player announces that she has the ace of spades. Now what is the probability that she has another ace? Gardner says it goes up--from 0.37 to 0.56. Weaver seems skeptical. After puzzling it over, I conclude that the statement "I have the ace of spades" is ambiguous in a way that "I have an ace" is not. The rule for saying "I have an ace" must be: if you have an ace, say so; if you have no ace, say nothing. Similarly, the rule for saying "I have the ace of spades" could be: if you have the ace of spades, say so; if you don't have the ace of spades, say nothing. I'll call this "rule 1." Then again the rule could be, for example: if you have no ace, say nothing; if you have one ace, say "I have the ace of [name the suit in which you hold the ace]"; and, if you have more than one ace, say "I have the ace of [name at random one of the suits in which you hold an ace]." I'll call this "rule 2."

Now, having set out all that, let's return to the bridge player who says "I have the ace of spades." I think Gardner is right, and that the probability she has a second ace is indeed 0.56, if the player has applied "rule 1." If on the other hand she says she has the ace of spades after applying "rule 2," I think the probability she has another ace is 0.37, just as it was after she said "I have an ace." What do you think?

Putting probability aside, this little episode, if I am right about how to analyze it, shows how much "disagreement" is really just linguistic confusion. Suppose a philosophy professor, having in mind the scope of Aristotle's system, declares

Aristotle was a catholic thinker

and one of her students, thinking the issue had to to with membership in the Church of Rome, replies

No, he was not!

It only seems that the professor and her student disagree about Aristotle, and in the same way it may only seem that Weaver and Gardner disagree about the solution to "the problem of the second ace."

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