# Boy Girl Paradox The paradox is formed via a pair of questions: > * Mr. Jones has two children. The older child is a girl. What is the probability that both children are girls? > * Mr. Smith has two children. At least one of them is a boy. What is the probability that both children are boys? A quick reaction may leave us thinking that the probability is 1/2 in both cases. However, looking at the below diagrams we see that is not the case:  The key idea here: > The **more specific information** we are given, the *more* we are able to **constrain our sample space**. In the case of knowing the second child is a girl, we are able to reduce our sample space to only two possible outcomes, of which there is a 50% chance of the other child being a girl. In the case of knowing at least one of the children is a boy (effectively an *OR* statement), we are only able to reduce our sample space down to 3 options, leaving us with 1/3 probability of the other child being a boy. This seems a bit strange since we are so used to viewing the events as independent: the sex of one child does not impact the other. *However*, from an *information perspective*, the outcomes are *not independent*! For instance, if we know that at least one of the children is a boy and I tell you the second child is a girl, we *know* the first must be a boy. So, given the totality of information present the sex of the first child was not independent, it was fully constrained given the information. Another helpful way to look at this is that *given no information* we would expect the probability of both children being boys to be 1/4. So, given this information that at least one is a boy, we update this probability (that both are boys) to 1/3. Looking at it this way is helpful since it feels more intuitive to be *increasing* the probability of the event (both children are boys) than decreasing it (it can feel like you are "decreasing" the probability since we tend to view the probability of being a boy as 1/2, and we are now saying the probability of this child being a boy is 1/3). --- Date: 20220309 Links to: Tags: References: * [Boy or Girl paradox - Wikipedia](https://en.wikipedia.org/wiki/Boy_or_Girl_paradox)