If a family has two children, what is the probability that at least one is a boy? If a family has two children, what is the probability that at least one is a boy, assuming there are only two sexes?
One way can be : The sample space is $\{BB, BG, GB, GG\}$ so that $P = \frac{1}{4}$.
This might also be a way : The sample space is $\{0 \ B, 1 \ B, 2 \ B\}$ so that $P = \frac{1}{3}$.
Are both models correct? Is there a fallacy? I usually see the answers to be $\frac{1}{4}$ so I am confused.
Edit : As I have been pointed out, I did mean $\frac{3}{4}$ and $\frac{2}{3}$.
 A: It's $\frac{3}{4}$.
The mistake in your second example
{0B, 1B, 2B}
is that these 3 possibilities are not equally probable.
I'm assuming that there's an equal probability of a child being a boy or a girl. With that in mind, if a family has 2 children then we have 4 possible outcomes each with equal probability:
GG
GB
BG
BB
Each of these outcomes has a probability of $\frac{1}{4}$. Now we can just count how many of them correspond to the outcome we're interested in, namely that at least one is a boy. The answer is clearly that 3 of them correspond to that, so the probability is $\frac{3}{4}$.
A: There are several other assumptions that must be made.

*

*Each child's gender is male or female with probability $1/2$, independent of all other children. That's certainly not quite true: slightly more boys than girls are born, while death rates for males tend to be higher. And I don't know if there is data on how independent they are.

*Decisions on whether to have another child are independent of the gender of the existing children.  Again, that's certainly not true.

