The response to Tuesday’s post, currently at 97 comments, has been very interesting. Since some of the commenters appear to be growing restless, I will put off until tomorrow my epic Iraq war post (based on my having recently waded through all 482 pages of Thomas Ricks’ subtly titled book Fiasco) and talk about probability instead.
Here’s the puzzle, in case you missed it the first time:
A shopkeeper says she has two new baby beagles to show you, but she doesn’t know whether they’re both male, both female, or one of each. You tell her that you want only a male, and she telephones the fellow who’s giving them a bath. “Is at least one a male?” she asks him. She receives a reply. “Yes!” she informs you with a smile. What is the probability that the other one is a male?
Reduced to its essence, the problem is asking this: “There are two dogs. One of them is male. What is the probability that the other one is also male?”
ANSWER: The probability is 1/3.
To see this, let us call one of the dogs X and the other Y. Initially we have four equally likely possibilities:
- X=Male, Y=Male
- X=Male, Y=Female
- X=Female, Y=Male
- X=Female, Y=Female
This is our starting point. If the question were simply, “What is the probability that two randomly chosen dogs are both males?” then the correct answer would be 1/4. (This would be a good time to state explicitly that we assume that males and females are equally likely.)
Now we receive information that allows us to update our probabilities. We learn that one of the dogs is a male. That means that option four now has a probability of zero. But this information does nothing to affect our assessments that the first three possibilities are equally likely. That is, the revelation that one of the dogs is a male is true regardless of which of the three scenarios we are in.
So we have three, equally likely possibilities, and in only one of them are both dogs male. So the answer is 1/3.
We could also approach this via Bayes’ Theorem. Let A=Both dogs are male and let B=One of the dogs is male. We seek P(A|B). (That is, we seek the probability that A has occurred, given that B is known to have occurred.)
According to Bayes Theorem, P(A|B) is equal to P(B|A) times P(A) divided by P(B).
- P(B|A) is 1. If we know both dogs are male, then it is a sure thing that one of the dogs is male.
- P(A) is 1/4. The sexes of the dogs are determined independently of one another. Since there is a probability of 1/2 that each dog individually is male, the probability that both are male is obtained by multiplying 1/2 by itself.
- P(B)=3/4. The only way for B not to occur is for both dogs to be female. Since that only happens 1/4 of the time, we see that B occurs 3/4 of the time.
Plugging this all in to Bayes’ Theorem gives us (1 times 1/4) divided by 3/4, which simplifies to 1/3.
A common incorrect answer is 1/2. The reasoning is that the probability that any given dog is male is 1/2, and the sexes of the two dogs are determined independently of one another. Therefore, knowing that one of the dogs is male effectively allows us to eliminate options three and four from our list. That leaves only the first two, equally likely, possibilities.
But there is a subtle, extra assumption being made in this argument. The reasoning above woud be valid if we were told specifically that dog X was male. If we ask something like, “If the dog on the chair is male, what is the probability that the dog standing by the door is also male?” then the answer is, indeed, 1/2. In that case we would be asking, in effect, “Does knowing that this particular dog over here is a male affect the probability that that dog over there is a male?” The answer is no. In this situation we are asking questions about the characteristics of individual dogs.
That is not the situation we have in this problem. In our case the question involves the characteristics of a pair of dogs. We are asking, in effect, “What is the probability that both dogs are male given that one of them is male?” The piece of information we receive from the shopkeeper does not relate to a specific, individual dog. Rather, it tells us something about the pair of dogs. By dividing the pair into two individual dogs, singling out one of them, and applying the shopkeeper’s information to that particular dog, we are making assumptions beyond what is given to us.
Another common incorrect answer is 1/4. Here the idea is that the probability that both dogs are male is “set in stone” so to speak. That is, the probability that two randomly chosen dogs are both male is 1/4, and simply being told that one of them really is male does nothing to change that fact. The error here is that knowing that one of the dogs is male means that the pair of dogs was not randomly chosen. Conclusions drawn about pairs of radnomly chosen dogs are therefore no longer applicable.
Here’s one more way of looking at it. Imagine that you flip two coins simultaneously. You do this multiple times, keeping track of the number of times you flip the coins. Every time you get two heads you make a little mark on a piece of paper. But here’s the catch: any time two tails come up you pretend like that flip never happened. You do not include it in your count of the number of times you flipped the coins. I am saying that if you do this a couple of hundred times, being careful to flip the coins fairly, then when you are done the number of marks on the page will be close to 1/3 of the number of tosses. Try it and see.
Browsing through the comments to the original post, I noticed a few challenges to the wording of the problem. One commenter suggested that the question was a bit misleading, since the term “the other one,&rdquo is vague. The idea is that to talk about “the other one” we must first single out one of the dogs, and that has not been done.
I don’t think this is correct, however. One of the dogs has been singled out, we simply don’t know which one. The person giving the dogs a bath has asserted that one of the dogs is male. If the shopkeeper asks him to prove it, he would point to one of the dogs and say, “See! That one is male.” Presumably that is what he did in answering the question in the first place. So the groomer has singled out one of the dogs by answering the question, we just don’t know which dog it was.
I would agree, however, that if you take the final question by itself, divorced from the rest of the problem, then you have a meaningless question. It makes no sense to talk about “the other one” unless you have a pair of things and one item in the pair has been singled out. I would suggest, however, that that is a rather bizarre way of reading the problem. In context I don’t think there is any ambiguity about what is intended. In the future, however, Ill take yet another commenter’s suggestion and phrase the question, “What is the probability that the other dog is also a male.”
Another commenter asked whether the dogwasher had seen more than one of the dogs. Actually, it’s irrelevant how many dogs the dogwasher saw. All that matters is that he has passed along to you the presumably correct piece of information that one of the dogs is a male.
Hopefully I’ve addressed most of the main threads that various people brought up. But if I haven’t, just let me know and perhaps we’ll do a third installment!