Hi all,Got the following question in an interview and due to the different answers i got from friends I would like your opinion too.In a hospital room for newborns, there are 2 boys and 3 girls. A woman gives birth to a baby and this is also added to the room. we randomly select 1 baby and it is a boy. What is the probability that the baby is a boy?Thanks in advance.

say the gender ratio at birth is 1:1 for boy and girl (more realistic number could be found here), then the prob that the new-born is a boy is (3/6)/(3/6+2/6)=0.6if you search on the forum, there are couple of old threads discussing similar problems (e.g., here)----- ----- ----- ----- -----not sure if this is the original source (problem 56 "babies in the nursery problem"), but as noticed by avishalom below, the girl # is irrelevant for the answer.

Hi Wileyw,and thank you for your quick reply! I have read the indicated thread and I would like to ask you if it is possible to tell me how you came up with this solution. The tree we have created is described as follows:Boys (2/6)Girls (3/6)Random (1/6) which continues to another tree with total pr(boy)=(1/6)*(1/2) and same for Pr(girl)=(1/6)*(1/2)Then calculated Pr(random|boy)=Pr(random&boy)/Pr(boy)=[(1/6)*(1/2)]/[2/6+(1/6)*(1/2)]=1/5Based on your answer, I believe you are using a different approach. I would really appreciate if you could tell me how you did it.Thanks

What wiley has reads: Pr(newborn is a boy|selected boy) = Pr(newborn is a boy & selected boy)/Pr(selected boy) = Pr(newborn is a boy & selected boy)/ (Pr(selected boy|newborn is a boy) + Pr(selected boy|newborn is a girl))

Hi Eddi,2 questions1) Shouldnt the denominator be (Pr(selected boy&newborn is a boy) + Pr(selected boy&newborn is a girl))? Otherwise you have to use something like this: P(B) = P(A&B) + P(A'&B) = P(B|A) P(A) + P(B|A')P(A') (ref. wikipedia)2) Could you please explain to me why the Pr(newborn is a boy & selected boy)=3/6? The random baby has a 1/2 probability to be a boy not 1. Sorry for asking all these questions just trying to understand how it works. maybe the tree i have in my mind is incomplete.Thanks for all the answers.

modraig, sorry about the sloppy notation. essentially this is a common application of Bayes' theorem (see e.g., here)say A = newborn is a boy, B = random selected baby (from 6) is a boy.as Eddi indicated, we want P(A|B), i.e., the posterior prob of A given B observed.Bayes' theorem: P(A|B) = P(B|A)*P(A)/P(B)P(A)=1/2 by assumption;if the newborn is a boy, there is P(B|A)=3/6 chance of selecting a boy.now P(B)=P(B|A)*P(A)+P(B|A')*P(A') as you said, where A' = newborn is a girl.similarly, P(A')=1/2 by assumption, and P(B|A')=2/6.hence the answer is (1/2)*(3/6) / [(1/2)*(3/6)+(1/2)*(2/6)] = 0.6(my previous post did not include these 1/2 factors)----- ----- ----- ----- -----your expression seems to answer the following question:given the randomly selected baby is a boy, what is the prob of him being the newborn one?

Cool so the original number of girls in the room does not matter at all. it cancels out. so if you originally had 0 girls or 1miliion girls doesn't matter.

Wow! that is a little counter-intuitive!

yep, the simplified formula is:p = P(boy born | boy selected) = (b + 1) / (2b + 1), where b is the number of boys initially.some special cases:b=0 -> p=1b->Inf -> p=0.5

Guys if you think rationally your answer must be wrong.If you make the probability tree and assume 50% chance of new born is boy or girl ( i.e. independant of the current population)then the probability you will get a boy randomly from the room cannot be more than 50%.Becuase a priori we have 2 boys and 3 girls making the probability of a boy be 2/5.After adjusting for the new born, in all possible configurations, the probability should be 5/12.Are you asking the right question? This to me is wrong.The tree is like thisNew born is boy (1/2) P(boy given new born is boy) 3/6 P(girl ..) 3/6NewBord is girl(1/2) P(boy ... is girl) 2/6 P(girl ...) 4/6given this space the total probability of boy is 5/6 / (4+3+5)/6

Can someone restate the question. Are we asking for the probability the baby born is a boy or probability of a random child from that ward being a boy?

QuoteOriginally posted by: marvasCan someone restate the question. Are we asking for the probability the baby born is a boy or probability of a random child from that ward being a boy?The probability that a random child picked from the ward is a boy, given that there are 2 boys, 3 girls and one unknown.

so i am right prob should be 5/12

QuoteOriginally posted by: marvasCan someone restate the question. Are we asking for the probability the baby born is a boy or probability of a random child from that ward being a boy?What is the probability that the baby born was a boy, given that a random child from that ward is a boy.In this context, the answer is 3/5 as mentioned earlier. edit: grammar