This post is a response to a comment on my March 3rd post titled, Logic and Probability Puzzle.
JeffJo, thanks for the comment. Throughout this response I will refer to the person giving the puzzle as “the puzzler” and I will use the notation P(X) to mean the probability of event X.
These are the assumptions needed to get to your answer of ½.
- Assumption 1: Each possible gender of a child is equally likely.
- Assumption 2: There are two possible versions of this puzzle:
- Boy version: “I have two children. One of them is a boy. What is the probability that I have two boys?”
- Girl version” “I have two children. One of them is a girl. What is the probability that I have two girls?”
- Assumption 3: Each possible version of the puzzle is equally likely if the puzzler is able to use either one because the puzzler has both a boy and a girl.
Using these assumptions, the following shows the percents of the universe of parents with two children by child combination and puzzle version:
| Boy Version | Girl Version | |
| Boy, Boy | 0.25 | 0 |
| Boy, Girl | 0.125 | 0.125 |
| Girl, Boy | 0.125 | 0.125 |
| Girl, Girl | 0 | 0.25 |
The probability of A given B is defined as: P(A|B) = P(A B)/P(B). In other words, the probability of the intersection of A and B divided by the probability of B. In other other words, the probability of both A and B occurring divided by the probability of B occurring.
P(the puzzler has two boys) given that the puzzler asked the boy version of the puzzles is P(the puzzler has two boys and asked the boy version of the puzzle) divided by the P(the puzzler asked the boy version of the puzzle). So, given the assumption listed above, we have:
(0.25)/(0.25+0.125+0.125) = (0.25)/(0.50) = 0.50 = 50% = ½.
I can see your point that it is necessary to know why we are given the information in a puzzle like this; however, I cannot accept a couple of these assumptions.
1) Assumption 3 is not reasonable. How can we know that each possible version of the puzzle is equally likely when the puzzler is able to use either one? A plausible argument can be made that the puzzler would be more likely to ask the boy version because the boy version has been told more often in the past which could influence the puzzler. This blog doesn't get a lot of views but the fact that the Skeptic’s Guide to the Universe used this puzzle could plausibly increase the likelihood of the boy version being asked by a puzzler with both a boy and a girl.
Assumption 1, each possible gender of a child is equally likely, is a reasonable assumption. This is common knowledge and while it might not be exactly 50/50, it is close enough to assume so in the context of a logic puzzle. Dissimilarly, the probability of a puzzler choosing the boy version over the girl version of the puzzle is not common knowledge and can only be determined by conducting a study. Such a study is likely outside the intended scope of a logic puzzle.
Lets scratch Assumption 3 and rework the puzzle using P(bv) to mean the probability of the boy version being chosen when the puzzler is able to use either version. We then have:
| Boy Version | Girl Version | |
| Boy, Boy | 0.25 | 0 |
| Boy, Girl | 0.25 * P(bv) | 0.25 * (1 - P(bv)) |
| Girl, Boy | 0.25 * P(bv) | 0.25 * (1 - P(bv)) |
| Girl, Girl | 0 | 0.25 |
(0.25)/(0.25+(0.25*P(bv)) + (0.25*P(bv))) = (0.25)/(0.25 + 0.50*P(bv)) = 1/(1+2*P(bv)). That is as simple as we can make the answer.
2) Assumption 2 is not reasonable either. How can we assume that there are only two versions of the puzzle? What about, “I have two children. One of them is a boy. What is the probability that I have two girls?” or “I have two children. They are both boys. What is the probability that I have two boys?” While it might seem silly to ask a version of the puzzle with an obvious answer, there must be some chance that the puzzler would ask these versions too. This fact impacts the puzzler with two boys also. So let’s try again using P(parent of two boys will ask the boy version) = P(X) and P(parent of both a boy and a girl will ask the boy version) = P(Y). Then we have:
| Boy Version | Girl Version | |
| Boy, Boy | P(X) | 0 |
| Boy, Girl | 0.25 * P(Y) | 0.25 * (1 - P(Y)) |
| Girl, Boy | 0.25 * P(Y) | 0.25 * (1 - P(Y)) |
| Girl, Girl | 0 | P(X) |
(0.25*P(X))/((0.25*P(X))+(0.25*P(Y)) +(0.25*P(Y))) = (0.25*P(X))/((0.25*P(X))+(0.50*P(Y))). That is the most simplified answer.
My Conclusions:
1) You argument is valid.
2) Your answer is incorrect because it uses unreasonable assumptions.
3) The result is a complicated answer that is not in-line with the intentions of the puzzle.
4) The puzzle should be restated to “Randomly select a person from the set of people who have two children at least one of which is a boy. What is the probability that that person has two boys?” The full version being, “Randomly select a person from the set of people who have two children at least one of which is a boy born on a Tuesday. What is the probability that the selected person has two boys?”
3 comments:
Thanks for the considered reply, Charles, but I must differ about your assessment of my assumptions:
Assumption 1: Each possible gender of a child is equally likely.
Correct. Everybody, including you, assumes this. They also assume gender is independent in siblings. It is called the Principle of Indifference: Say two options exist that are indistinguishable, except by name. If no reason is given why either one should be more, or less, probable than the other, then you can only assume they are equally probable.
Assumption 2: There are two possible versions of this puzzle.
No, I don’t assume that (well, it is unequivocally true that there are two versions. But I don’t assume anybody formulates the other one, which I think is what you meant.) I again used the Principle of Indifference to assume that if person has one boy and one girl, that (s)he is equally likely to tell me about the boy or a girl. AS A CONSEQUENCE of that assumption, if anybody WERE to formulate the "girl version," it would have to have the same answer. But if the assumption I actually made is false, the two must have different answers.
Assumption 3: "Each possible version of the puzzle is equally likely..." I make no such assumption.
Now, let's talk about the assumptions you made.
Assumption 1: Each possible gender of a child is equally likely, and gender is independent in siblings.
Like I said before, this is a universal assumption.
Assumption 2: If a randomly-selected parent of one boy and one girl is going to tell you information about only one of his or her children, it will always be the boy and will never be the girl.
This assumption is absurd. Not absurd like where you claim that it is impossible for us to know which version of the puzzle a parent of a boy and a girl would choose. I agree with that - but the reason behind why I agree is precisely why the Principle of Indifference applies. Your assumption is absurd because you not only are assuming what you claimed is impossible to know, you are assuming it is perfectly imbalanced in one direction based on the observation you argued it was impossible to predict.
Essentially, you are assuming that the proposition "I will tell you I have a boy IF AND ONLY IF I have a boy" is a true statement. And you can't justify that, because it simply isn't true.
Hi JeffJo. Thanks again for your comments.
To be clear, I am conceding that the answer to the puzzle, as originally stated in my first post, is not 2/3. I find your broader argument to be valid which has led me to this conclusion. As such, I am not making the assumptions you describe in your assessment of my Assumption 2. My point was rather that the possible puzzle versions when a person has a boy and girl is not like the probability of a child being a boy or a girl, which is observably 1/2. I’ll add that the Principle of Indifference does not apply because the possibilities are not exhaustive and, more importantly, are distinguishable my more than name alone. Thus I conclude that the answer is not definitively 1/2 or 2/3, but a formula with probabilities represented by variables.
I’ll further explain my point about the inapplicability of the Principle of Indifference which I believe is the only thing we disagree on. In your first comment on my first post you say, “And if you don't know how the decision [to choose a version when you have a boy and a girl] was made, you can only assume a random choice”. I’m saying that if you don't know how the decision was made, you have to assign a variable for its probability. If I was forced to guess what the probability was I would guess somewhere around 55% of the time the boy version would be chosen because of cultural biases towards favoring boys over girls; however, I don’t have much confidence in that guess. It’s not like flipping a symmetric coin where we have zero knowledge of anything that could influence the probability of heads or tails.
The original puzzle statement, with an option to indicate the more, or less, complex version was: "I have two children. One of them is a boy (born on a Tuesday). What is the probability that I have two boys?"
The issue is NOT how to re-interpret that statement so that the answer you want to be correct, is correct. The issue is how to interpret it in the first place.
Simple version: "I have two children. One of them is a boy. What is the probability that I have two boys?" There are two contested answers: 1/3 and 1/2. In order to get 1/3, you have to assume that any parent of one boy and one girl, who desires to tell you about only one child, will always tell you about the boy. That it is the fundamental nature of any parent of two to ignore daughters. It is an unreasonable assumption; not because of the Principle of Indifference, but because it assumes universal adherence to this assumption. If there exists even one such parent, anywhere in the world, who would tell you about a girl, the assumption is false.
In order to get 1/2, you have to assume the Principle of Indifference. That doesn’t mean that you assume no such parent of two will be biased. It means that the problem statement provides you with no information about how this particular parent might be biased. It means that you assign not just a "variable" to the possible bias; it is a "random variable" which, by necessity, must have a symmetric distribution. So P(bias=X)=P(bias=1-X). And when you integrate over that, you get the same answer as assuming the probability is 1/2 to begin with - because you can't get a different answer for the symmetric problem.
Post a Comment