As a child is boy or girl; this doesn't depend on it's elder siblings. So the answer must be 1/2, but I found that the answer is 3/4. What's wrong with my reasoning? Here in the question it is not stated that the couple has exactly 4 children
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2Please explain how you arrived at $3/4$ as an answer. – hmakholm left over Monica Jan 21 '17 at 20:27
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I see this answer in that book where I saw the problem but I can't understand how it is 3/4 – Bivas Das Jan 21 '17 at 20:29
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1I can imagine an interpretation of the question where the correct answer is $4/5$, but $3/4$ has me stumped. – hmakholm left over Monica Jan 21 '17 at 20:30
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2it must be $\dfrac{1}{2}$ in my view – Kiran Jan 21 '17 at 20:32
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What book gives the answer $\frac 34$? – lulu Jan 21 '17 at 20:34
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I also think so – Bivas Das Jan 21 '17 at 20:35
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Outlines of statistics by Gun, Gupta,Dasgupta – Bivas Das Jan 21 '17 at 20:36
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I'd say the natural answer was $\frac 45$ but it depends on how you interpret the phrasing. – lulu Jan 21 '17 at 20:36
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@lulu, how 4/5? – Kiran Jan 21 '17 at 20:38
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In question it is clearly stated that first 3 children are girls so sample space:{GGGG,GGGB} probability 1/2 – Bivas Das Jan 21 '17 at 20:38
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1@Kiran let's say, as is natural, that all combinations for the $4$ kids are equiprobable. There are $5$ combinations of which it's true that three are $G$, namely $GGGG,GGGB,GGBG,GBGG,BGGG$, and in four of those five cases the fourth kid is $B$. – lulu Jan 21 '17 at 20:40
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1@BivasDas That's not how the question you posted reads, nor is it how the duplicate question was phrased. Please post the question exactly as it appears with no alterations. – lulu Jan 21 '17 at 20:41
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@lulu, thanks for the point. – Kiran Jan 21 '17 at 20:41
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@BivasDas All I have is the question as you phrased it, and nothing is said about "first three". If you meant to write something else, please edit your post. – lulu Jan 21 '17 at 20:43
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@BivasDas, me too share the same view. consider a coin is tossed 5 times and we get head in the first 4 tosses. then what is the probability for getting head in 5th toss? it is $\frac{1}{2}$ in the normal way of interpretation. same case here i believe. – Kiran Jan 21 '17 at 20:44
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@lulu u'r point is correct – Bivas Das Jan 21 '17 at 20:44
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As a (possibly irrelevant) remark: I can give an interpretation which yields $\frac 15$. This would be the case if we applied Laplace's rule of Succession. That is, assuming we know that $G,B$ are both possible, we assume independence, we observe $GGG$ for the first three, but do not assume that the probability for each birth is $\frac 12$. Rather we assume that the probability, $p$, is chosen uniformly at random (but is the same for each birth).
Of course, as the comments show, there are other sensible interpretations which yield $\frac 12$, or $\frac 45$. I haven't come up with one that yields $\frac 34$ but perhaps I lack imagination. Worth noting that the solutions which give $\frac 12$ or $\frac 45$ both assume that there are exactly $4$ kids. If you don't want to make that assumption...well, then you need to make some other assumption about the meaning of the question or you can just fall back on Laplace (really a last resort, I'd have thought).
The main point here is that phrasing and interpretation are critical. If the text presents the problem without clarification then I'd say the reader was obliged to go through the various sensible readings and either answer several of them or, at a minimum, clearly indicate what assumptions they are declaring.
lulu
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You will get $1/4$ when you know that the family has four children and three girls. That means only three girls and not four girl. The cases are BGGG, GBGG, GGBG, GGGB. So the probability that fourth (youngest) child is son is $1/4$. You may get $3/4$ when the question is the probability that the youngest child is girl – Archisman Panigrahi Jan 22 '17 at 12:34
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1@ArchismanPanigrahi Right. so if the interpretation is "there are exactly $4$ kids and exactly $3$ of them are girls then find the probability that the fourth child is a boy" then you get $\frac 14$. Here, the OP specifically says that we aren't to assume that there are exactly $4$ kids, but it isn't clear how we are to proceed in that case. – lulu Jan 22 '17 at 13:55