Boy Or Girl Paradox/ Two Child Problem.

Abstract: The following article is a comprehensive analysis of the Girl or Boy Paradox also called the two-child problem. This article explains reasons behind why the paradox is counter intuitive, to give

a rational thought and explain the paradox by the use of mathematical tool for a proper analysis.

Introduction

The Boy or Girl paradox surrounds a set of questions in probability theory, it is also known as The Two Child Problem, Mr. Smith's Children and the Mrs. Smith Problem. Gary Foshee, at the Ninth Gathering 4 Gardner Conference in March, 2010, presented a probabilistic puzzle, the solution of which was quite counter-intuitive. It generated intensive discussion on the internet, with some intriguing contributions, and some described as misleading. According to this problem, we consider a family having two children and that one of the two children is a boy born on a Tuesday, and are asked “what is the probability that there are two boys?” On first acquaintance, the information about the day of birth may seem to be irrelevant and cannot affect the result. But things are not so straightforward. Foshee presented an answer that astounded his audience and that appeared to defy intuition. The initial formulation of the question dates back to at least 1959, when it was featured in Scientific American by Martin Gardner in his "Mathematical Games column." He called it The Two-child Problem, and phrased the paradox as:

• Mr. Jones has two children. The older child is a girl. What is the probability of both the children being girls?
• Mr. Smith has two children. At least one of them is a boy. What is the probability of both the children being boys? (No mention of Tuesdays here).

Gardner initially gave the answers 1/2 and 1/3, respectively, but later acknowledged the ambiguity of the second question. Its answer might be 1/2, depending on what information was available beyond a boy being the only child. The ambiguity, depending on the exact wording and possible assumptions, was confirmed by Bar-Hillel and Falk, and Nickerson.

The paradox has spurred much controversy. A lot of people argued strongly with a lot of confidence for both sides, sometimes showing disdain for those who took the opposing view. The paradox arises from whether the setup of problems for the two questions is similar. Its intuitive response is 1/2. This answer is intuitive if the question leads the reader to believe that the second child's sex (i.e., boy and girl) has two equally likely possibilities, and that the likelihood of those results is absolute, not conditional.

Other variants of this question, with varying degrees of ambiguity, have been popularized. by Ask Marilyn in Parade Magazine, John Tierney of The New York Times,and Leonard Mlodinow in Drunkard's Walk.

Common assumptions.

The two possible answers share a number of assumptions. First, it is assumed that the space of all possible events can be easily enumerated, providing an extensional definition of outcomes: {BB, BG, GB, GG}. This notation indicates that there are four possible combinations of children, labelling boys B and girls G, and using the first letter to represent the older child. Second, it is assumed that these outcomes are equally probable.This implies the following model, a Bernoulli process with p = 1/2:

1. Each child has the same chance of being male as of being female.
2. The sex of the children is independent of the sex of the each other.
3. Each day of the week is equally probable, likewise each month, each star-sign, etc.

(These assumptions can be challenged, but we are not concerned here with genetic subtleties, chronological quirks or astrological aberrations).

The two-child problem. 

Under stated conditions what is the probability that a ‘two child family’ there are two boys?

Considering the simplest problem, if there are two children what will be the probability of two boys?

There are four possible cases,

Ω = {BB, BG, GB, GG}, where for example BG signifies that the first-born child in the family is a boy and the second is a girl.

Each of the possible four outcomes are equally likely, i.e., all of these four outcomes have the same chance of occurring. Thus, we can assign equal weights to all of them. The probability that two boys occur is P= ¼

Now if we consider the problem that the first-born child is a boy. The sample space, 

Ω = {BB, BG}. As most outcomes are equally likely, the probability of two boys is P=½

For the next problem we consider that one of the children is a boy. The sample space will be 

 Ω = {BB, BG, GB}. here the probability of two boys is P=⅓. This problem was popularized by Martin Gardner. It is mostly known as the two child Paradox. The intuitive problem is P=½, whereas the mathematical reasoning leads us to the answer P=⅓.

Tuesday’s Child

Introducing a dichotomy. We assume that all the children fall into one of the two categories denoted by the subscript 1 and 2, with relative frequencies L and M, where N=L+M.  We also assume that the frequencies are independent of the sex of the child. There are now 16 possibilities in the configuration of the two-child family which can be denoted as 

We have indicated the relative frequencies in the bracket.

The total weight is 4N2, where N2 is the total for each of the four 2×2blocks. Using this array to address Problem 3 (in which at least one child is a boy), we must consider the sample space comprising all cases except those in the bottom right-hand 2 × 2 block. We find immediately that,

 as before.

Consider next the probability of two boys given that one child is a boy in Category 1, i.e., that B1occurs. The total weight is 4LN − L2. The weights for the three cases having two boys sum to 2LN −L2. Thus, the probability is

where p=L/N is a relative frequency of category 1. Now taking to limits,

Thus, the smaller L is, compared to N, higher is the probability of a two-boy family given that condition. Now let us at least one of the children is a boy born on a Tuesday. Then L = 1 and N = 7 so P = 1/7. Thus, by (3), the probability of two boys is P=13/27

The surprise here that the fact that a child being born on a Tuesday have any sort of influence on the result

 Analysis of Ambiguity

It certainly seems surprising that the weekday of birth of one child can influence the probability of the sex of the other. The fact that we actually use the information of a boy being born on a Tuesday at the outset to determine the sample space is the critical factor. We consider the question: "For all two-child families for which at least one child is a boy born on a Tuesday, what fraction of those families have two boys?   "In order to simplify matters, let us return to the problem of Gardner: 'Among all two-child families for which at least one child is a boy (born on any day of the week), for which fraction of these families are two boys? We have found above that the answer is P = 1/3

Even if the boy is born on a Tuesday the chance of the other child being born as a boy is still 50:50.

Bayes’ Theorem

In the context of Bayes’ theorem, let us consider the probability that there are two boys within a two-girl child family.

However, the sample space for a two-child family is {BB, GG, BG, GB}.

By H the hypothesis X belongs to {BB} is denoted.

A further condition is then introduced K, which refer to the condition that at least one child is a boy.

Therefore, according to Bayes’ Theorem –

PH|K=PHPHP(K)

P (H) = 1/4 is the unconditional probability of H.

Also, we can logically say that P (K|H) =1, since two boys also refer to the condition ‘at least one boy’. 

Now we will take into consideration two child families with one or more boys. Since according to the sample space there are three such cases, P (K) =3/4.

Putting all the determined values in the Bayes Theorem equation we get, P (H|K) = 1/3.

Conclusion.

Psychological Investigation –

A relevant question in the context of statistics, very often does not have a specific answer. However, the boy-girl problem is exempted from this fact because in this case ambiguity is not the determinant of how the intuitive probability is calculated.

With regard to this Savant created a survey according to which he says that most people intuitively arrive at the answer ½ in Gardner’s problem rather than 1/3. This phenomenon makes this an interesting study for many psychologists and psychological researchers.

Two such people Fox and Levav created an experiment to test out how people estimate conditional probability –

• Mr. Smith says: “I have two children and at least one of them is a boy”. With this given statement people were asked about the probability of the other child being a boy as well.
• Mr. Smith says: “I have two children and it is not the case that they are both girls”. With this given statement people were asked about the probability of Mr Smith having two boys.

Interestingly enough, it was so observed that in the first case people have a tendency to assume that there are only two possible outcomes for the other child. The second one on the other hand provides an indication to people that there are four possible outcomes out of which 1 has been eliminated. This interesting observation can be further substantiated by saying that out of people asked this question 85% replied ½ to be the answer to the first question while 35% people answered the same to the second question.

References

Boy or Girl paradox. (2020, August 03). Retrieved August 05, 2020, from https://en.wikipedia.org/wiki/Boy_or_Girl_paradox

Lynch, P. (2011). The Two-Child Paradox: Dichotomy and Ambiguity. Retrieved August 05, 2020, from https://www.maths.tcd.ie/pub/ims/bull67/2011-6-1.pdf

Marian, J. (n.d.). The ‘day of the week boy or girl’ paradox explained. Retrieved August 02, 2020, from https://jakubmarian.com/the-day-of-the-week-boy-or-girl-paradox-explained/