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 $48.5\%$.
- 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.