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BGonline.org Forums
Tricky probability puzzle
Posted By: Timothy Chow In Response To: Tricky probability puzzle (Jason Lee)
Date: Sunday, 29 October 2023, at 1:03 a.m.
There is no doubt that conditional probability can be counterintuitive. However, in the specific case of the puzzle you mentioned (ace versus ace of spades), part of the blame can be laid on the textbooks, which often gloss over subtle questions about mathematical modeling of the real world in their eagerness to get to the mathematical formulas.
For example, when you look at your cards and say, "One of my cards is an ace," textbooks often jump to conclusions about the process by which you decided to make your statement. The typical assumption is that your rule of behavior was:
1. If at least one of my cards is an ace, then I will say, "One of my cards is an ace."
2. If neither of my cards is an ace, then I will say, "Neither of my cards is an ace" (or maybe I will say nothing).
While this assumption isn't unreasonable, it's not the only possible assumption. Maybe your rule of behavior was instead, I'll choose one of the two cards at random, look at it, and if it's an ace, I'll say, "One of my cards is an ace"; otherwise I'll say nothing, even if it turns out that the other card is an ace. This rule of behavior leads to a different probability calculation. You might complaint that this assumption is far less plausible than the other assumption, but the point is that it is perfectly consistent with the information we are given. (And I would argue that the latter scenario isn't entirely absurd in the real world.) So an additional assumption has been slipped in without comment. In the real world, it's very important to be cognizant of the assumptions that are made when applying probability theory to analyze empirical data, so textbooks do students a disservice by failing to call attention to these assumptions.
Getting back to the ace versus the ace of spades, we see that in order to answer the question about the ace of spades, we have to introduce some assumption about the rule of behavior that led to the declaration. The rule can't be exactly the same as before, because under the previous rule, you never said anything about the suit. Moreover, it's no longer even obvious what your rule is. Do you look at your cards, pick one at random, and say, "One of my cards is the X of Y" whatever X and Y happen to be? Or do you look at your cards, and if at least one of them is the ace of spades, then you say, "One of my cards is the ace of spades" and otherwise you say nothing? Or do you adopt some other rule? Different assumptions can lead to different answers. Once we spell out all the assumptions, it becomes less surprising that two similar-sounding problems could lead to different answers.
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