Is this some kind of mathematician’s joke? Intuitively, 4/7 can not be right. If you start out with 3 boys and 500 girls, 4/7 is not right. Likewise, 1 boy and 1 girl, etc. Without talking about the probabilitiyes, can anyone explain this in laymans’ terms? Is this one of those instances where you are proving that a probability analysis can not be applied to real life situations? I am befuddled.
if you start with zero boys and 1 girl, and they bring you a boy - what’s the probability that your wife gave birth to a boy? it is one evidently. if you start zero boys and 1,000,000 girls, and they brought you a boy - you still know the probability that your wife gave birth to a boy is one. it should pretty clear that in this scenario, the probability is one based on your knowledge that you start with zero boys, and is completely independent on how many girls you start with. why do you think that suddenly when you start with say, 3 boys instead of zero boys as above, the probability will start to depend on the number of girls? they girls are irrelevant here.
CardShark Wrote: ------------------------------------------------------- > The Baby Nursery > > > > A nursery currently has 3 boys and x girls in it. > > > Your wife if rushed to the hospital and gives > birth to a baby. The baby is now added to the > nursery. > > I go to the nursery and ask a nurse to randomly > bring me a baby. She does, and it happens to be a > baby boy. I think the stuff before is just meant to distract and the trick has to do with just focusing on the final question: > What is the probability that your wife gave birth > to a boy? This is 50%.
sublimity Wrote: ------------------------------------------------------- > > What is the probability that your wife gave > birth > > to a boy? > > This is 50%. That’s what I was thinking. Almost seems like the riddle about the man going to St. Ives. Lots of extra info but if you are assuming it is 50/50 in your probability analysis, then isn’t the probability that she had a boy 50/50 regardless of the makeup of the nursery?
hezagenius Wrote: ------------------------------------------------------- > That’s what I was thinking. Almost seems like the > riddle about the man going to St. Ives. Lots of > extra info but if you are assuming it is 50/50 in > your probability analysis, then isn’t the > probability that she had a boy 50/50 regardless of > the makeup of the nursery? 4/7 is the conditional probability… P(delivers boy/brings boy), that’s what the question seems to be asking for.
^^No, because you are given extra information beyond just the simple fact that she gave birth. You are also given the information that this child was placed in a nursery with 3 boys and x girls And also the information that randomly selecting a child from this sample yields an observation of boy (hence the odds are slightly better than 50-50 that your wife gave birth to a boy). You have to update your beliefs based on this information to arrive at the best probability measure, otherwise you are not using all available information to efficiently form your posterior beliefs. Make sense?
^ the crux of the matter is whether we interpret the final question as a conditional probability (making use of all the information) or an unconditional probability (not making use of all the information). so there are two possible interpretations here.
adavydov7 Wrote: ------------------------------------------------------- > hence the odds are slightly better than 50-50 This is a good intuitive way to look at it. I’ll provide a finance analogy: A stock goes up $1 half the time, and down $1 half the time. Given it was up $1 on Friday, is the probability the stock is up for the week greater than 0?
^true, but when would you ever rationally make a decision without incorporating all available information? (in a mathematical or economical world at least since we clearly do this in our daily lives because we are irrational as a species).
I love MILF’s.
Ahahahaha, awesome username. Did you create it solely for the purposes of this thread?? ROFL!!! How long before it gets banned? Odds?
Think about it this way. There are two possibilities: 1) The hospital has N+1 boys and x girls 2) The hospital has N boys and x+1 girls Given that a random baby I choose is a boy, which scenario is more likely, 1) or 2)? Intuitively, you would choose 1), since there are more boys in this scenario. In other words, randomly selecting a boy has given you more information and the probabilities are no longer 50/50. How different is the biased probability from 50/50? That depends on how many boys there were to begin with. If you know that there were few boys, then the bias from 50/50 will be large. If there are many boys, then the bias will be small. The number of girls doesn’t matter since, you are finding the conditional probabilities of scenarios 1) and 2) given that you have *already* chosen a boy in the sample. Does this make sense?
adavydov7 Wrote: ------------------------------------------------------- > ^true, but when would you ever rationally make a > decision without incorporating all available > information? (in a mathematical or economical > world at least since we clearly do this in our > daily lives because we are irrational as a > species). one way to tackle a difficult and complex problem is to assume away any relevant information and simplify the problem to an elementary one with a known and obvious solution. this approach works very well in situations when you are clueless. typically, one applies this strategy via backwards induction. you need to start with an answer that you know, and proceed to interpret the question in a manner that agrees with your solution by carefully ignoring any given information that might be contradictory.
Ok, I guess I simply don’t believe that a single sampling is giving me real data, hence I’m going to ignore it. This is based on practical understanding and not strict mathematical reasoning. I’m assuming that I know the stochastic likelyhood is 50-50 at the start. I know there are both boys and girls in the selection pool already based on the data. Somebody pulls a boy, since I know it was already possible and I’m not putting any weight on a sample size of 1, I’m going to stick to the stochastic fundamentals of 50-50. I know concretely that is the biological probability and want to resist being mislead by a single pick. I dunno, I am weak in math and would like to go get a masters in it for this very reason.
Where did this question come from and what did the book/professor/website say the answer was? I would have picked 50% rather than 57% if it was multiple choice.
This reminds me of a funny problem about mathematical problems vs. reality. Say you flip a coin and it lands heads 99 times in a row. What is the probability that it is heads the 100th time? Of course, probabilistically it is 50%. However, there’s a close to 100% chance it’s a loaded coin, a more realistic possibility than the 99 heads events of p = (1/2)^99. So therefore, the real, street smarts, financial market smarts answer is 100% is the probability that it will come up heads the next toss after those 99 tails.
in this case though, i’d say the real “street smart” would be to ignore the single sampling whereas in the coin situation, you have 99 flips. So I guess a more in depth analysis comes down to weighting of data. In other words, which data do you wait in a given scenario.
^ yeah, and it may have to do with subjective concerns as well. to get a bit deeper, it is a problem about the relation between epistemology (what you know, a statement about your subjective ignorance) and ontology (what is, a statement about objective reality). there’s a meta level of analysis that needs to be done, to put the problem into context and to assign to the problem “meta-” probabilities, about what the relative weighting, probability, or importance is of epistemological and ontological concerns. : )
4/7 is not the ans. to the question on the probability of a boy is born by the wife. As many pointed out above it is simply counterintuitive in extreme scenarios (x = zero or million baby girls). 4/7 however is the answer on the probability of a boy bought to you by the nurse on the condition that it is 50/50 for your wife to deliver a boy or girl, which IS THE QUESTION ITSELF. My simple answer to this is inconclusive. It looks like there is not information sufficient to draw a conclusion. In mathematical terms: event A = a boy is delivered from the Mrs. event B = a baby boy is bought to you by the nurse we are seeking P(A), which mathmatically is = P (A & B) / P (B|A)? P (B|A) = 4/(4+x) P (A & B) = P(A) x P(B) IF they are independent, but in this case they are NOT. As such, i believe no conclusion on the probability that a boy is born can be drawn. my 2 cents
I love this forum, we took a simple math brain teaser and turned it into a discussion involving: 1. The validity of the 50-50 boy-girl assumption given historically observed biological trends 2. An embroiled discussion regarding the validity of the second data set (independent vs dependent probability) 3. Decided weighting the data assumptions may be the way to go (epistimology vs ontology) 4. And finally concluded that it was inconclusive