Transcription
If I had two children and all you knew was that one of them is a boy born on a Tuesday, then what are the chances that both children are boys? Okay, I know what you're thinking. What does being born on a Tuesday have to do with anything? Surely I must have included that detail just to try to throw you off or something, right?
Well, let's take a look at the original version of this puzzle first, which was posed by none other than Martin Gardner over half a century ago. Here's how the problem goes: If I have two children and at least one of them is a boy, then what's the probability that both children are boys?
Yep, this is the exact same problem minus the born on a Tuesday part. And the solution is pretty straightforward. See, having two children creates four possible outcomes, and knowing at least one child is a boy eliminates one of them, leaving us with three equally probable cases. Now, since only one of these cases involves both children being boys, then the probability that I have two boys must be one in three, or about 33%.
All right, cool. Let's go back to the initial problem now. We know the answer would just be 1/3 if the born on a Tuesday part were ignored. So, the real question is, would knowing that extra detail actually affect the probability? Or better yet, can knowing any extra information about the boy affect the probability?
Well, suppose if I had instead told you that the older of my two children is a boy. In this case, notice that this would rule out the second column entirely, leaving us with only two possibilities. So, the probability that both children are boys must be 50%, which means that extra detail we added caused the answer to jump from 1/3 to 1/2.
But, here's where it gets even stranger. It turns out that mentioning any distinguishing feature bumps the probability up from 1/3 to 1/2. Yep, we get the same result if I, for example, tell you the taller child is a boy. Or the one with darker hair is a boy. Or even the one who prefers pizza as a boy.
All right, that should settle it then. The answer to the puzzle must be 50%, except it actually isn't. And that's because being born on a Tuesday isn't exactly a distinguishing feature. Since for all we know, both children could have been born on a Tuesday. So, mentioning it doesn't uniquely distinguish one child from the other.
Okay, uh, where do we go from here then? Well, we know that each child can be one of two genders and can be born of one of seven days of the week, giving us 14 equally probable cases. This means that each two-child family belongs to one of 196 equally probable cases. After restricting ourselves to those with at least one boy born on a Tuesday, we're left with 27 potential cases. Now, notice only 13 of these involve both children being boys. Therefore, the answer to the puzzle is 13 over 27, or about 48%.
Okay, now it's your turn. What if instead of mentioning the day of the week, I tell you the month in which my son was born? What's the probability that both children are boys this time?