I think it is 1/2, because the second part does not affects the first part.
3 girls and 2 boys are in a room one more child is added , nurse picks a kid randomly and it is a boy
What is the probability that the added child is boy?
This problem is either too hard or too easy.
Can you tell me what is the correct answer?
Thank you.
It's fairly easy.
Assuming it is equally likely for the added child to be a boy or to be a girl.
Probability that nurse adds a boy = 1/2
Probability that nurse adds a girl = 1/2
Since we haven't been told which child was added (that is why we are assigning probabilities above), we have to account for both the possibilities of nurse adding a boy and nurse adding a girl.
By basic application of Bayes Theorem and the law of total probability, we can find that
Probability of selecting a boy = (Prob of selecting a boy given that added child was a girl)*(Probability of adding a girl child) + (Prob of selecting a boy given that added child was a boy)*(Probability of adding a boy child)
After having calculated the probability of selecting a boy, we proceed to calculating the probability of added child being a boy given picked child is boy.
Note that situation is inverse that of the one in which we calculated the probability of selecting a boy.
Using the conditional probability formula,
Probability of the boy being the added child given the picked child was a boy = Probability of the boy being the added child and the boy being the picked child / Probability of picking a boy
The Probability of the boy being the added child and the boy being the picked child can be found by multiplying the probability of picking the boy given the added child was a boy and the probability of adding a boy child.
I hope the method is clear!