Two-Cent Probability

Probability is undeniably one of the most counterintuitive areas of mathematics. What makes it especially “dangerous” is that, at first glance, it seems simple, almost obvious. If we set aside Measure Theory, sigma-algebras, and other fairly complex mathematical objects unfamiliar to most people, the question “What is the probability that a given event happens?” is generally clear and undeniably useful. But there is a reason casinos are always full: our minds are very easily fooled by hard, cold numbers, especially when they clash with our expectations.

Tuesday’s Child

Even though I like to think I am fairly comfortable with probability by now, this post that I stumbled across on Reddit sent me into a mini crisis.

Mary has two children, and she tells you that one of them is a boy born on a Tuesday. What is the probability that the other child is a girl?

My instinct immediately said the answer had to be $50\%$, assuming a $50\%$ chance of having either a boy or a girl and a uniform distribution of birth days across the week. But in the comments, many people were quite convinced the correct answer was $\frac{14}{27}$, which made me wonder whether I was missing something.

Kids as Coin Tosses

To get to the bottom of this, it helps to simplify the problem by turning it into a two-coin toss, where each coin has a $50\%$ chance of landing heads or tails. To make the analysis clearer, we can list all possible outcomes in a table:

In other words, there are four possible outcomes, each with probability $25\%$.

The Hidden Twist in the Wording

When tackling a math problem, you always need to pay close attention to wording, because that is often where the trick is hiding. The original statement is a bit ambiguous, so let’s rephrase it like this:

After tossing two coins, if the first is heads, what is the probability that the second is tails?

From the table, we can see that if the first coin is heads, only two outcomes are possible: (H, H) and (H, T). So the probability that the second coin is tails is $\frac{1}{2}$, or $50\%$.

But the question could also be phrased this way:

What is the probability that, by tossing two coins, I get one head and one tail?

Again, from the table, we see there are two outcomes out of four that satisfy this condition: (H, T) and (T, H). So the probability remains $\frac{1}{2}$, or $50\%$.

Toss two coins and, assuming there is at least one heads, what is the probability that there is also at least one tails result?

Now the context changes completely. In this case, we only consider outcomes that contain at least one heads: (H, H), (H, T), and (T, H). Among these three combinations, only two include tails: (H, T) and (T, H). So the probability of also having tails, given that there is at least one heads, is $\frac{2}{3}$, about $66.67\%$.

What is the probability that, after tossing two coins, after learning there is at least one heads, there is also at least one tails result?

In this case, the information itself is part of the probability update. The probability that the event “at least one heads” occurs is $\frac{3}{4}$, since it rules out (T, T) and leaves only (H, H), (H, T), and (T, H). From that point onward, we can proceed exactly as in the previous case: among those three outcomes, the probability of having tails is $\frac{2}{3}$. Multiplying the two gives $\frac{3}{4} \times \frac{2}{3} = \frac{1}{2}$, or $50\%$.

In short, everything depends on which set of outcomes we are considering, and on whether we treat the clue as a probabilistic event or as information that rules out some possibilities.

Conclusion

Going back to the original problem, the answer depends on how we interpret the question. To understand our interpretation, we just need to ask: “What would Mary have said if she did not have a son born on a Tuesday?” If the answer is “She would have said something different,” then the correct probability is $\frac{1}{2}$. If instead the answer is “Mary’s statement rules that possibility out,” then the correct probability is $\frac{14}{27}$.