History and Background
Confusing English is "probably" the most encountered aspect of beginning probability theory questions. Being extremely precise with wording in such problems is crucial, as even using exact English with modern mathematical notation can lead to ambiguity. The easiness of which one can slip into philosophizing what exactly defines "chance" or "randomness" is illustrated in a pivotal work Probability, Statistics, and Truth by the famous Richard von Mises circa 1928. Professor von Mises lays down a purely frequentist view stating:
Richard von Mises (courtesy of Wikipedia)
“The rational concept of probability [...] applies only to problems in which either the same event repeats itself again and again, or a great number of uniform elements are involved at the same time.”
A contrarian stance is seen in Bayesian thought, which considers events as worthy of having chance without the baggage of being thought of as a draw from a repeated action, instead using prior beliefs about an event along with "new" info to assess probability. Philosophical difficulties over the definition of probability compound even further in situations involving conditional probability, with the power to ruin all intuition for problems and even their solutions. The Boy or Girl paradox is one such problem which highlights a confusing thought process intersecting English phrasing, seemingly obvious biological ideas, and generative processes. The problem was originally posed as a pair of plainly stated questions by the master of aggravating puzzles, Martin Gardner [2]. The major concept we need from here is of course Bayes' Theorem; below the notation P(A|B) refers to the probability of event A given that event B occurred.
The Problem Statement
Martin Gardner, puzzle master. Courtesy of Wikipedia
“Q1: Mr. Smith has two children. Given that one of the children is a boy, what is the probability that both children are boys?”
“Q2: Mr. Smith has two children. We meet him walking along the street with a young boy whom he proudly introduces as his son. What is the probability that Mr. Smith’s other child is also a boy?”
Martin Gardner gave the answer to Q1 as 1/3 and the answer to Q2 as 1/2; the paradox is subsequently that Q1 and Q2 seemingly state and ask exactly the same thing but some reasoning can give two different answers. The controversy ignited by this discrepancy was contentious and sparked a wide array of debate and commentary, where I tried to highlight key writings in my references listed at the end. Despite this, I think the paradox is strictly solved when making clear assumptions about how exactly the information stated in the problem was obtained. Here I hope to write down the counter-intuitive solution as clearly as possible.
Q1 Solution (the What)
First, there are only 4 options for families of two as far as the sex: {GG, GB, BG, BB} where B,G are boy and girl respectively. As we know there is at least one boy in the family, the only options now are {GB,BG,BB}; that is to say there are not two girls. Each child is assumed to be picked by Mr. Smith with equal chance, so each configuration of family is equally likely. We are interested in BB, so the answer is 1/3 since we "picked" a family configuration as a whole at random from all such families.
Q2 Solution (the What)
Again there are 4 options for family configurations, {GG,GB,BG,BB}. Mr. Smith himself shows us his son. We know there is one other child and we are to determine the sex of this child. Assuming that both sexes are equally likely, with the logic that clearly sibling sexes are independent, we have that the other child has a 1/2 chance to be boy since we are "picking" one child instead of a family configuration. In other words, we are considering two equiprobable and independent draws from the pool {G,B}.
The Q1 Generating Process (the How)
The logic is sound in both solutions above, however each approach makes unspoken assumptions about the nature of the provided information [5]. The difference in the above solutions, you might have caught, is exactly how we are given the information about the observed child. In the following analysis we define k_1 and k_2 as kid 1 and kid 2, the event BB is as before, the word "draw" refers to existence of k_1 and k_2 as realizations of a random variable whereas the use of "observe" refers to finding out what that realization is.
The Q2 Generating Process (the How)
The different phrasing and situation of finding out one of the kids' sex of Q2 is broken down into its randomization process:
The act of getting the sex information completely from Mr. Smith, who knows the sex of both of his children, completes one entire draw on the {G,B} pool.
Conclusions
The paradox is that both analyses above are valid given sufficiently vague English phrasings of Q1 and Q2. A psychological study into how people think about conditional probability presented what was shown to be a clearer version of Q1 [4]:
- Mr. Smith says: 'I have two children and at least one of them is a boy.' Given this information, what is the probability that the other child is a boy?"
- Mr. Smith says: 'I have two children and it is not the case that they are both girls.' Given this information, what is the probability that both children are boys?"
Number 1 above led people down the path of the Q2 process, it is the biological heuristic that the sex of the children are independent creates a jumping of rationale to the answer of 1/2, i.e the sampling procedure includes some mistaken "order" of the children. The clarified phrasing of number 2 above makes it clear to restrict the sample space appropriately to {BB,BG,GB}.
In the original writing of the problem, Gardner recognized the nature of this paradox and made it clear that "failure to specify the randomizing procedure" is responsible for the confusion. The problem itself then becomes a cautionary tale for statisticians and those reading the results of their work, that it is not just what, but how.
References
- Wikipedia
- Martin Gardner (1954). The Second Scientific American Book of Mathematical Puzzles and Diversions. Simon & Schuster.
- Grinstead and Snell’s Introduction to Probability, Available for free here
- Craig R. Fox & Jonathan Levav (2004). Partition–Edit–Count: Naive Extensional Reasoning in Judgment of Conditional Probability
- Stephen Marks and Gary Smith (2010) The Two-Child Paradox Reborn? Available here.