Mathematics
The Probability Paradox: The Mind-Boggling Two-Child Problem Explained
An intriguing enigma that guarantees to deliver surprises
Understanding probability can be challenging, even for experienced math students. When I was first introduced to the concept, it seemed like nothing but fun — rolling dice or flipping coins. But as I delved deeper into probability theory, it quickly became clear just how complex and daunting it could be.
Even so, the many fascinating and counterintuitive results from probability theory never cease to inspire me. The unique amusement and amazement it brings set it apart from other math topics and I believe many would agree with me.
In this article, I will be discussing the Two-Child Problem in probability theory, a deceptively simple problem ready to reveal surprising insights.
Also referred to as the Boy or Girl Paradox, Mr. Smith’s Children, or the Mrs. Smith Problem, the Two-Child Problem was formulated by Martin Gardner[1]. The problem consists of two parts:
Question A: Mr. Jones has two children. The older child is a girl. What is the probability that both children are girls?
Question B: Mr. Smith has two children. At least one of them is a boy. What is the probability that both children are boys?
While the answer to question A is widely agreed to be 1/2, there is debate surrounding the answer to question B. Gardner first provided an answer of 1/3, thus being different from that of question A. This may have surprised some readers. Later on, Gardner acknowledged that question B can be interpreted in multiple ways and that both 1/2 and 1/3 are acceptable answers.
What does it mean by “can be interpreted in multiple ways”? You will be surprised how far this question will take you.
“Probability” is a difficult concept
To proceed, let us re-write question B here:
Mr. Smith has two children. At least one of them is a boy. What is the probability that both children are boys?
To some smart people, the statement “At least one of them is a boy” obviously contains hidden ambiguity. They will ask: how does one obtain such information?
Does how you find out that information matter?
Consider the following two different scenarios:
Scenario 1 — All families with exactly two children, where at least one child is a boy, have gathered and formed a group. We randomly pick one family from the group, and that family turns out to be the Smiths.
We declare “Mr. Smith has two children. At least one of them is a boy.” and ask “What is the probability that both children are boys?”. The declared information is fully consistent with our knowledge. We have not disclosed how Mr. Smith was randomly selected from a group of interest because we believe such information is not important.
In this scenario, simple counting is all it takes to find the answer. The possible gender combinations of the two Smith children are BB, BG, and GB. Assuming them to be equally likely, the probability of finding BB is 1/3.
Scenario 2 — All families of exactly two children have gathered and formed a group. We randomly pick one child out of the group, the child turns out to be a boy and his father is Mr. Smith.
Again, we declare “Mr. Smith has two children. At least one of them is a boy.” and ask “What is the probability that both children are boys?”. As in the first scenario, the declared information is fully consistent with our knowledge. We have not disclosed how the boy was randomly selected from a group of interest because we believe such information is not important.
In this scenario, one can use the concept of conditional probability to find the answer:
P(BB|b)=P(BB)/P(b)
- P(b) is the probability of finding the randomly selected child to be a boy, which is 1/2
- P(BB) is the probability that the Smith’s children are both boys, which is 1/4, based on the assumption that BB, BG, GB, and GG are equally likely
- P(BB|b) is the probability of both siblings being boys given that the randomly selected child is a boy
Putting everything in, one gets P(BB|b)=1/2, which is different from Scenario 1.
Strange, isn’t it?
Probability as frequency
In both Scenario 1 and 2 above, I have associated Mr. Smith, or the boy, with a particular group of interest from which a random selection is performed. And I have shown that modifying the underlying group of interest, or how the random selection is performed will lead to a different answer.
The above approach to interpreting “probability” is in line with the frequentist perspective.
Frequentist, as the name suggests, interprets “probability” as the relative frequency of an event occurring, given infinite repetitions of an experiment. This interpretation is no stranger to most of us. I was taught this way when young and it has been the only interpretation and understanding of “probability” for me for decades. It may be the same for you.
When one interprets “probability” as the frequency of events, one must take into account the setting where one’s experiment is conducted. In question B of the Two-Child Problem, different settings generate different answers, and so without the disclosure of the setting, one could not proceed.
Probability as a measure of certainty
In some situations, the frequentist interpretation of “probability” does not offer much help.
Consider this modified Two-Child Problem I made up:
I am a detective.
I have a murder case at hand, all I know is: the murderer is a father of two sons.
I have two suspects, Mr. Jones, and Mr. Smith. I have no prior knowledge about them.
One day, I received two secret tips.
Tip 1: Mr. Jones has two children, the older child is a boy.
Tip 2: Mr. Smith has two children, at least one child is a boy.
Due to the urgency of the case and limited resources, I must set priorities and make a quick decision on whether to first investigate Mr. Jones or Mr. Smith. Can you help?
Clearly, in this situation, I am concerning myself with “certainty”: I have two similar pieces of information, and I need to tell which of them carries more “weight”, or delivers more “certainty”. I do not concern myself with how the secret tips were obtained, and simply take them in as if “a little bird told me”.
To me, it is logical to make my decision between Mr. Jones or Mr. Smith this way:
There are only 4 possible gender combinations for the Jones and Smith children: BB BG GB GG
Tip 1: It allows me to narrow down to 2 possibilities: BB BG.
Tip 2: It allows me to narrow down to 3 possibilities: BG GB BB.
Because I am looking for a father of two sons, I conclude that Tip 1 delivers more “certainty” to me and I would first investigate Mr. Jones.
In the same line of thought, one could go back to the original Two-Child Problem and claim that the answer to question B is 1/3.
This way of thinking is in line with the Bayesian interpretation of “probability”.
Thomas Bayes was an 18th-century English mathematician, best known for developing Bayes’ theorem which is a fundamental principle of modern probability theory and statistical inference.
In Bayes’ view, “probability” represents the degree of certainty, confidence, etc. Even in situations where there is no die-throwing, no coin flipping, or any similar long-run process involved, one can still develop a concept of “probability” as a measure of decision quality or serves as a logical guidance for decision making.
I know someone with a Bayesian mindset who sticks to 1/3 being the only answer to question B of the Two-Child Problem, he is my husband. To him, the two questions A and B are put together on purpose to illustrate the different “weights” of two pieces of highly similar information. He tends to conclude that only through the Bayesian lens will one find a meaningful message behind the Two-Child Problem.
Seemingly basic, the Two-Child Problem touches on the very fundamental aspects of probability theory and more. If you find the paradox mind-boggling, you are definitely not alone.
In most math curricula, the topic of “probability” is introduced in middle school but one must never say: “probability” is easy stuff. “Probability” is a complex and sophisticated concept and I hope the paradox of the Two-Child Problem has given you a glimpse of its true complexity.
[1] Martin Gardner (1961). The Second Scientific American Book of Mathematical Puzzles and Diversions. Simon & Schuster. ISBN 978–0–226–28253–4.
