What is the probability that three babies are boys, given that at least one is a boy? I know this is a simple problem but I am arguing with a friend about its solution, so I want to show him an "official" proof!
Suppose that in any birth, the probability to have a boy is $48.5\%$.

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*If we have three persons expecting to deliver, what is the probability that at least one of them gives birth to a boy?

*If we know that at least one will give birth to a boy (suppose we have accurate ultra-sound results), what is the probability all three will have a boy?

For the first question, we calculate the probability of one NOT having a boy, which is $1-0.485 = 0.515$ and then the required probability of all three not having a boy is $0.515^3 = 0.1365$ so the probability that at least one will have a boy is $1-0.1365 = 0.8634 = 86.34\%$.
For the second question, since the three events are independent, the probability that all three will have a boy given that at least one will have a boy is equal to the probability that the other two will have a boy. Is it $0.485^2$? I am not sure about the second one.

In the second problem, please note your set excludes the case where none of the fetus is a baby boy as you know at least one of them is.
So the probability in the second case that all three fetuses are baby boys $ \displaystyle = \frac{0.485^3}{1-0.515^3}$
