In this article, I will attempt to use a single probability question to explore a wide range of social, psychological, pedagogical and mathematical ideas in a very short space of time. The question is this:
A woman has 2 children. One of them is a girl. What is the probability that her other child is a boy?
Feel free to pause here and consider this question on its own for a while if you wish to.
I may have told a small lie just now. It is possible that there is a second probability question also to be considered:
A woman has 2 children. One of them is named Sue. What is the probability that her other child is a boy?
But this depends, of course, on whether this is a different question mathematically or whether, to all intents and purposes, “one of them is a girl” and “one of them is called Sue” amount to the same thing.
Now, we mathematicians live in a bit of a strange universe. We usually deal with pure, abstract concepts, yet we often can’t resist the temptation to cast our ideas in the context of so-called “real world” scenarios. Ahmed has 91 watermelons; Jane has 257 tangerines. Really?
The question (or questions?) above possibly fit this profile. There is an element of trust involved; trust that the reader will make some assumptions about the situation, and indeed about the world, in the spirit of mathematical simplicity. These might include…
- Boys and girls are all there is.
- Each gender is born with equal and independent probability.
- Boys are never called Sue.
Jonny Cash may disagree with the final bullet point, but on a more serious note, the discussion of the first bullet point has potential social and contemporary political implications. While a teacher of, say, English may be well versed in the management of discussion of controversial topics within the classroom, this is something that Maths teachers are often inclined to avoid. But we must be prepared to consider the implications of questions and statements that imply a binary and immutable world of sex and gender. We might want to brush such questions aside in the name of mathematical simplicity, but our more socially conscious students might not let us!
The second bullet point is also a simplification. Historically, the ratio of male-to-female births worldwide has been about 105:100. Interestingly, the ratio of males to females amongst people of any age is closer to 102:100, reflecting the longer life expectancy of females.
How often do discussions like this happen within your maths classroom? Do you encourage them or view them as an unwelcome distraction? With the growing popularity of Core Maths qualifications, which focus far more on real-world interpretation and debate, this is an issue that many teachers may need to confront in the near future.
But enough of all that! Let’s get back to some nice simple maths, shall we? For the original question, having made the three assumptions above, you may have considered what is sometimes referred to as a “sample space”. There are four possible outcomes for 1st child – 2nd child; these are boy-boy, boy-girl… this is the point, when attempting to read maths, that my brain often starts to hurt. So, I will stop here and draw you a nice picture instead:
We are equally likely to be in any of the quadrants, except that we can’t be in the boy-boy one because we know one of the children is a girl. So, the answer is \(\frac2 3\) then, right?
But what about the “other” question? Or did you even consider it to be a different question at all? Psychologists identify a phenomenon known as “substitution”, where, when faced with a very difficult question, we unconsciously substitute it for one which sounds “similar” but which we can more easily answer. Have you done this here? And has this led to an error? Surely this substitution would only be completely safe if
Is named Sue \(\iff\) Is a girl
But this is clearly not the case. In fact, Jonny Cash songs and other complications aside,
Is named Sue \(\implies\) Is a girl
In other words, all Sues are girls, but not all girls are Sues!
Does this matter, though? Well, actually, yes, it really does. Consider this picture, which represents a sample space for 36 women, all of whom have two children. The girls named Sue are coloured green:
We have equally as many Sues in the girl-girl quadrant as in the two mixed-sex quadrants combined. This means that if one girl is named Sue, the probability that her sibling is a boy is \(\frac 1 2\) a different answer from the first question!
What new assumption did we make here, though? Although you may not have noticed it, we have assumed that:
- The name Sue is randomly given to newborn girls with a fixed and independent probability.
Is this assumption reasonable?
What if the name Sue is like Marmite? Parents either love it or hate it. Love of the name Sue is distributed at random amongst the people giving names to children.
If this is the case, parents who love the name Sue will always name their first girl Sue, and those who hate it will never name a child Sue. The picture would then look more like this:
Within each quadrant with any girls in at all, an equal portion of families have a Sue. So, the probability that Sue’s sibling is a boy is now back to \(\frac2 3\)
So, what are we to make of this? Assume the name Sue is given out at random, and we get one answer \(\frac1 2\). Assume it is like Marmite to parents, and we get an answer of \(\frac2 3\). Unfortunately, we need to confront the real world again. Who knows what the true “marmitivity” of the name Sue is?
The actual answer to the question must, I am sure, lie somewhere between \(\frac1 2\) and \(\frac2 3\), but without real data, I am as clueless as anyone else as to where!
by Michael Gibson