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This is similar to the two child problem.

So there are five cases, the gender of the children written according to their age,

  1. GGGG
  2. GGGB
  3. GGBG
  4. GBGG
  5. BGGG

So the probability should be $\frac{4}{5}$. But the given answer is $\frac{3}{4}$.

Am I missing something?

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Archisman Panigrahi
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2 Answers2

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The problem statement says that a family has 4 kids, and that 3 of them are girls. Because they don't say "at least" 3 of them are girls, we understand they what they actually mean is "exactly" 3 of them are girls and the remaining kid is a boy. What is the probability that the $4$th child is a boy?

By $4$th child they mean the youngest. So how are the genders distributed among ages? We can have either of the $4$:

  • GGGB;
  • GGBG;
  • GBGG;
  • BGGG;

where the leftmost is the older sibling and the rightmost is the younger one. Thus, out of those $4$ possible arrangements, only in one the youngest kid is a boy and thus the answer should be $\frac14$. $\frac34$ is the answer to "what is the probability that the 4th kid is a girl" or "what is the probability that the boy is not the 4th kid" or anything along those lines.

RGS
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    Nice way to get to $\frac{1}{4}$! Can we think of a feasible scenario that gets us to $\frac{3}{4}$? – Bram28 Jan 21 '17 at 16:19
  • @ArchismanPanigrahi it would be better phrased that way, yes, if we assume your friend gave you the wrong answer by mistake, 1 minus the right answer actually, thus making the right answer $\frac14$. For the answer to be $\frac34$ it is more likely that your friend forgot a word when giving you the problem statement. – RGS Jan 21 '17 at 16:23
  • @Bram28 I find it more likely that his friend made a mistake when giving him the problem statement than there being a situation where the right answer actually is $\frac34$ - given the information we have as of now – RGS Jan 21 '17 at 16:24
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    I agree. Which shows another important aspect of these kinds of probability questions: as stated in English, questions about probabilitycan often be interpreted in many different ways. – Bram28 Jan 21 '17 at 16:27
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In a real life situation, it depends on how you know that the family has 3 girls. Here are two different scenarios:

A. You run into the mother with 3 of her children with her that are all girls, and she tells you that she has a 4th child. Now the chance of the 4th child being a boy is $\frac{1}{2}$

B. You are in a room with a bunch of parents, and someone asks: 'who is a parent of exactly 4 children, at least 3 of which are girls?', and the mother of this family raises her hand. Now the probability of her 4th child being a boy is $\frac{4}{5}$, following exactly your explanation.

There may be other scenarios yet, leading to different answers yet, and some might even lead to a probability of $\frac{3}{4}$, but I think scenarios isomorph to either one of these two cases are most likely to happen in real life. So I don't like the answer of the book either.

Bram28
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  • I hate dubious probability questions :P – RGS Jan 21 '17 at 16:20
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    @RSerrao I kind of like them! :) It really urges us to think real clear about the assumptions we are making when we are doing math, and whether they apply to a real life situation. – Bram28 Jan 21 '17 at 16:21