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I've found this question on a book and I'd like a review in my answer.

The king comes from a family of 2 children. What is the probability that the other child is his sister?

srodriguex
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    The question may be ambiguous as stated: see https://en.wikipedia.org/wiki/Boy_or_Girl_paradox#Second_question. Under one possible interpretation it seems to me that it might depend on whether the ruler of the country is sometimes a queen, or is always a king, which would be an interesting twist. – Trevor Wilson Oct 10 '13 at 01:28

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First of all, take note about @Micah comment and suppose the male-preference primogeniture is observed.

This is a Conditional Probability question.

We'll use the followin notation:

$P(A|B)$ means the probability of event $A$ occurs given that event $B$ has occurred.

By definition on page 59 of Sheldon Ross book, we know that $ P(A|B) = \dfrac{P(AB)}{P(B)}$, where $AB = A\cap B $. Another notation is $|X|$ that means the number of elements in set $X$.

As the king comes from a family of two children, we are given two tips: (1) This family has a boy, the king. (2) The king has a sibling. What we want to know is the probability this sibling be a girl. In other words, what's the probability of the two children be each one of each gender.

Let B the event of possible children where at least one is a boy, the king. So $B=\{(b,b), (b,g), (g,b)\}$, where $(x,y)$ means the gender of each child and the possible values are $b$ for boy and $g$ for girl. Then $A$ is the event that the king's sibling is a girl, $A=\{(b,g),(g,b)\}$.

The sample space $S$ contains all possible outcomes, $S=\{(b,b),(g,b),(b,g),(g,g)\}$.

It follows that $AB=A$ and $|AB|=|A|=2$, $|B|=3$ and $|S|=4$.

So, we have:

$$P(A|B) = \dfrac{P(AB)}{P(B)}=\dfrac{\dfrac{|AB|}{|S|}}{\dfrac{|B|}{|S|}}=\dfrac{|AB|}{|B|} = \dfrac{2}{3}$$

The end.

srodriguex
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    Looks good. Note that this answer is only valid in countries with male-preference primogeniture. In countries such as Sweden with absolute primogeniture, the king can't have an older sister: that is, $(g,b)$ is not a possible outcome, so the probability is $1/2$. – Micah Oct 10 '13 at 01:26
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    It seems to me that your interpretation of the problem would be valid if all you know about the randomly chosen family of two children is that at least one of the children is a male (and you want to find the probability that there is also a female.) But now what if you add the extra information that one of the boys is a king? Doesn't that make it more likely that there are two boys, because kings are fairly rare and a family with two boys is more likely to contain a king than a family with one boy? – Trevor Wilson Oct 10 '13 at 02:35
  • @Micah, although you've aroused an interesting fact, I don't see why it would affect the outcomes since nothing there is no mention about succession, but the fact that the king can have an older sister. – srodriguex Oct 10 '13 at 02:42
  • @srodriguex: The point is that if the country uses strict primogeniture, the outcome $\langle g,b\rangle$ is not in $A$ or $B$: the king is necessarily the older child. That changes the probability to $\frac12$. Your solution assumes that the country uses male primogeniture, so that the boy will be king even if he’s younger than his sister. – Brian M. Scott Oct 10 '13 at 03:42
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    @Trevor: No. As far as gender distribution goes, one of these siblings is a king tells you no more and no less than at least one of these siblings is male (in the absence of empirical data to the contrary, of course: it’s conceivable that the distribution of biological gender among royal offspring is different from that of the population at large). – Brian M. Scott Oct 10 '13 at 03:50
  • @Brian I must confess that I am confused then. The event "at least one sibling is a king" is the conjunction of the events (1) "at least one sibling is male" and (2) "at least one of the male siblings is a king", so learning it should have the same effect as learning (1) and then learning (2). It seems that, given (1), learning (2) should raise the probability that both siblings are male, because, again given (1), event (2) has a positive correlation with the event "both siblings are male." What am I missing? – Trevor Wilson Oct 10 '13 at 03:55
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    @TrevorWilson: Event (2) does not have a positive correlation with the event "both siblings are male". If you are the king, your younger brother definitely is not... – Micah Oct 10 '13 at 04:01
  • @Trevor: I don’t know, because I can’t see why you think that (2) should raise the probability that both siblings are male. – Brian M. Scott Oct 10 '13 at 04:01
  • @Brian Given a randomly selected family with 2 children, doesn't learning that two of them are boys (instead of just one) raise the probability that one of them is king? (Just like learning that one of them is a boy instead of zero would raise the probability that one of them is king.) And then we use the fact that correlation is a symmetric relation. – Trevor Wilson Oct 10 '13 at 04:06
  • @Trevor: Sure, but that has nothing to do with this problem. You haven’t learned that one is king independently of learning that one is a boy: you know that one is a boy because you know that one is a king. – Brian M. Scott Oct 10 '13 at 04:09
  • @Brian I'm afraid I'm still confused, and I don't know quite what you mean by "independently" and "because" in this context. Isn't the effect of learning a conjunction the same as the effect of learning one conjunct and then learning the other? – Trevor Wilson Oct 10 '13 at 04:15
  • @Trevor: There isn’t really a conjunction here. We were told only that one sibling was a king. We then inferred (on the basis of the normal meaning of king) that at least one sibling was male; that information was already contained in the original datum. It’s a conjunction $p\land q$ where we already know that $p\to q$. – Brian M. Scott Oct 10 '13 at 04:20
  • @Brian I know it's only a conjunction in a trivial sense. However, am I not free to rewrite the event "one child is a king" as an equivalent conjunction, and then substitute one equivalent event for another in any place I desire? – Trevor Wilson Oct 10 '13 at 04:23
  • @Trevor: Yes, if you don’t then use it as the basis for an incorrect argument! If strict primogenture is in force, one sibling is king tells you no more (of relevance) than the eldest sibling is male. If male primogeniture is in force, one sibling is king tells you no more (of relevance) than at least one sibling is male. The additional information contained in king merely reduces the relevant population; it does not change the distribution of biological genders (again, barring empirical data to the contrary). – Brian M. Scott Oct 10 '13 at 04:36
  • @Brian I have laid out my argument in detail over the course of these comments, so if it is incorrect then perhaps you can point to an incorrect step? It's time for me to go home now, but I will think more about this tomorrow. Meanwhile, thank you very much for your patience. – Trevor Wilson Oct 10 '13 at 04:41
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    @Brian I think I understand now. It seems like, under your assumptions, you have (a) every family with at least one male is equally likely to contain a king, whereas I am assuming (b) every male is equally likely to be a king. To argue for (a) vs. (b) I think we'd have to formalize more properties of "king" besides just being a type of male. For example, if "king" were replaced with "male lottery winner" then (b) would seem more correct. Perhaps with the actual meaning of king, (a) is more correct. In any case, I am no longer concerned by our disagreement. – Trevor Wilson Oct 10 '13 at 05:49
  • @Trevor: Under male primogeniture it’s definitely false that every male is equally likely to be a king, and I am not assuming that every family with at least one male is equally likely to contain a king. I am assuming that the distribution of biological gender amongst siblings is the same for pairs of siblings that include a king as it is for pairs of siblings in general. – Brian M. Scott Oct 10 '13 at 05:56
  • @Trevor: Male vs. strict (or absolute) primogeniture and their effects on the problem have been mentioned right from the beginning. Yes, I did state that assumption before. Twice. See my comments about empirical data to the contrary. And it’s irrelevant whether you add containing at least one male: that excludes only the girl-girl case, which isn’t at issue anyway. I was in fact making the more general assumption, though the argument requires only the narrower one. – Brian M. Scott Oct 10 '13 at 06:12
  • @Micah Now I fully realized your observation. My answer is not fully correct. So I ask you if you can answer this question so as to I can check it as the right one. – srodriguex Oct 27 '13 at 12:32
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As Martin Gardner points out regarding the similar "boy or girl" paradox (see here), a failure to specify the randomizing procedure could lead readers to interpret the question in several distinct ways. (This wording is partly Gardner's and partly the Wikipedia article author's.)

For example, the following alternative to srodriguex's interpretation also seems reasonable:

We randomly select a king. Then we are informed that the king has exactly one sibling. What is the probability that the sibling is female?

In this interpretation the answer is 1/2, not 2/3. One way to see this is by symmetry, because the fact that kings are male does not affect the calculation in this interpretation.

Trevor Wilson
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  • "@Trevor Wilson" I copied this question from the book I mention. The question is on Chapter 3, wich covers Conditional Probabilities. The book gives the answer 2/3, but don't give the arguments. In my first try I reasoned 1/2 and when I saw this wasn't the answer, I created the arguments I posted here to reach 2/3, and this is why I posted here, to see if it's correct. – srodriguex Oct 10 '13 at 02:51
  • @srodriguex Yes, and my answer is that it's not unambiguously correct, and seems to depend on the interpretation. Even with your interpretation, I'm not totally convinced yet that the answer is 2/3. – Trevor Wilson Oct 10 '13 at 03:46
  • @TrevorWilson One question. I think that the answer 2/3 is wrong because the king is either the first boy or the second boy. If the king is the second boy the options (b,g) and (g,g) can be eliminated. Similar idea can be apply if the king is the first boy. What do you think? – Beginner Jun 06 '18 at 20:26
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    @Beginner Sorry to not respond earlier. I think we are meant to assume that it is the firstborn son who becomes king (see the reference to male-preference primogeniture in srodriguex's answer.) As a math question it's a bit of a trick question because it's not clear which real-world facts about kings we are meant to use. – Trevor Wilson Jun 15 '18 at 22:31
  • @TrevorWilson Thank you for the explanation. Do not worry about the delay. – Beginner Jun 17 '18 at 00:46
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It is unambiguously correct ( assuming odds boy/girl 50-50 & male always king). Only possible family structures: b,b / b,g / g,b - all family structures are equally likely ; therefore probability King having a Sister 2/3 (b/g ; g/b).

On the below : " We randomly select a king. Then we are informed that the king has exactly one sibling. What is the probability that the sibling is female?" 1/2 - 1/2 ; But this is not a different interpretation of the same question ; It is a different question !! Because if you randomously select a King, the odds of the King coming from a family with 2 boys rather than a family with one boy, is two times as big. ( and because we know there are two times more families with one boy and one girl ; we get >> b,g/ g,b/ b,b/ b,b AND it is 50-50) ; or said in a different manner ; now it is possible that the King has an older brother - if the younger brothers would be excluded from the random drawing - it would be 2/3-1/3 again.

Few notes for fun : * Are the odds Boy/Girl 50-50 ? * King could be one of identical twins - meaning that there are ( small) odds in favour of b,b ; making the likelihood a little lower ( non identical twins assume normal distribution ; and identical girl twins are not in the possible poule anyway)

Cheers,

Sytse

Sytse
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Reduced Sample space is {bb, bg, gb}. Are equally likely if assumption is made that elder son will be king. But if anyone can be king bb, bg, gb are not equally likely but P(bb) = 1/2

Sample space should be changed if we assume there is no discrimination with boy or girl and on the basis of elder or younger.

{KB, KG, BK, GK} that's reduced Sample space according to given condition. Without the given condition that ruler was king or queen. P(Male) = 1/2 is assumed P(Elder is ruler) = 1/2 is assumed Sample space is {KB, BK, KG, BQ, GK, QB, QG, GQ}

K means male who is king Q is female who is Queen}

P(sibling is female) can be solved by considering. Sample space = {KB, BK, KG, GK} Event = {KG, GK}

P = 1/2

Mahendra Goyal
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