I ran into an article the other day in Science News called When intuition and math probably look wrong, which led me to an article in Wikipedia on the The Two-Child Paradox. I was bothered by the quality of the Wikipedia article, and when I followed up the citations, they all pretty much said the same thing, derived from the same few sources, with very little proof: "The correct answer is 1/3, and if you think otherwise it's because you're a victim of understandable but incorrect intuition." What follows is my proof that this is in fact a false paradox, unlike, say the Monty Hall Paradox, which, while nicely less-than-intuitive, is still mathematically correct.
What gives this "paradox" its paradoxicalness is the magical P(BB)=1/3, and I propose that it is simply an incorrect result caused by the legendary Martin Gardner performing an illegitimate transformation on the discrete probability distribution representing the possible states of a base system to produce one for a transformed system. The transformation is rendered illegitimate by the subtle character of certain conditional probabilities.
Let's call the initial system "predict a parent's 2 kids", or P2K. It consists of the following set of assumptions.
If we care about birth order we get the following distribution.
| GG | 0.25 |
| GB | 0.25 |
| BG | 0.25 |
| BB | 0.25 |
Now we append a NEW assumption to P2K.
Martin innocently prunes GG off the tree and arrives at P(BB)=1/3.
But let's look at example 4.1 in the conditional probability section of a publication cited in the Wikipedia article, Grinstead and Snell's Introduction to Probability:
An experiment consists of rolling a die once. Let X be the outcome. Let F be the event {X = 6}, and let E be the event {X > 4}. [...] P(F) = 1/6. Now suppose that the die is rolled and we are told that the event E has occurred. This leaves only two possible outcomes: 5 and 6. In the absence of any other information, we would still regard these outcomes to be equally likely, so the probability of F becomes 1/2, making P(F|E) = 1/2.
Simple tree-pruning is only guaranteed to work "in the absence of any other information", but we HAVE other information! Saying that one of the children is a boy implies P(GG)=0, but it ALSO implies THE EXISTENCE OF THE BOY, who now can and must participate in 4 equally probable possible states! This transforms the system itself in a fundamental way, requiring a complete recalculation of the probability distribution. We can't just simply prune the tree; we can't assume the existence of a boy when we discard GG and ignore it when we preserve the relative probability of BB; we can't simultaneously know and not know something.
Let's call the new system "predict a parent's 2 kids given at least one boy", or P2KG1B; call our boy theBoyWeKnowAbout, or simply theBoy; and calculate a new probability distribution to represent P2KG1B (again, if we care about birth order):
| bigSister - theBoy | 0.25 |
| theBoy - littleSister | 0.25 |
| bigBrother - theBoy | 0.25 |
| theBoy - littleBrother | 0.25 |
Note that this new distribution exactly represents the possible states of P2KG1B, as is required.
Now, if we want to use the same terms as P2K's to describe the new distribution, we can:
| GB | 0.25 |
| BG | 0.25 |
| BB | 0.5 |
It's not about foolish attachment to intuition. The SAME system transformation (coming to know of theBoy) that causes P(GG)=0 also causes P(BB)=0.5, because P2KG1B has 2 distinct possible states (each of probability 0.25) that can be labeled BB, whereas P2K has only one.
We also can easily ignore birth order, yielding the following distributions.
P2K:
| girls | 0.25 |
| boys | 0.25 |
| mixed | 0.5 |
P2KG1B:
| boys | 0.5 |
| mixed | 0.5 |
No semantic ambiguity, no 0.333..., no mystery, no psychology, no beagles, no paradox: just a subtle mistake.