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Assume that every couple can only have exactly 1 child or two children, with those outcomes being equally likely. Ignore any silly extra factors (e.g. 1 child dying, another being alive).

If I choose a human on the face of the earth randomly, what is the probability that they had a sibling?

As a follow-up, what would be the general answer if couples had $i$ children with probability $\pi_i$?

My thought for the simple case is either 1/2 or 2/3, and I can think of compelling reasons for both.

Siblings get counted twice, and only-children get counted once, leading to 2/3.

A given person's parents were equally likely to give him a sibling or not, leading to 1/2.

This kind of problem occurred to me one day. A similar problem occurs to me when I think about divorce rates. (e.g. if the divorce rate is 50%, is the probability a person is divorced 50% or 66.6%?) Any help is appreciated!

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    $\begingroup$ $\frac{2}{3}$ is the correct answer. We were picking a person uniformly at random from among all people to ask if they had a sibling or not. We are not picking a parent uniformly at random and looking at a child of theirs and asking if they have a sibling or not. $\endgroup$
    – JMoravitz
    Jan 15, 2021 at 23:17
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    $\begingroup$ Imagine this... you have one bag with a single white ball in it. You have another bag with $100000000$ red balls in it. Dumping the balls out and mixing and then picking... you are much more likely to pick a red ball than the single white. Picking a bag first and then picking one from the bag... you'll be just as likely to get white as red. That is effectively what the difference between your approaches is here. They are totally different questions since the way we pick is different. The way the question is asked, it is clear that we have dumped out the balls and mixed before picking. $\endgroup$
    – JMoravitz
    Jan 15, 2021 at 23:19

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The answer is 2/3 and more generally the probability of having $i$ siblings is:

$$\frac{i\pi_i}{\sum_{k} k\pi_k}$$

where $\pi_i$ is the probability of a couple having $i$ kids.

One way to see this is assume there are 100 couples. Then we would expect 50 of them to have 1 kid and 50 to have 2 kids so we have 150 kids total and 100 siblings which is 2/3. You can do a similar calculation to see why the general equation holds.

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  • $\begingroup$ +1 The inclusion of an example is a great way to make these things clear! $\endgroup$
    – Carl Schildkraut
    Jan 15, 2021 at 23:37
  • $\begingroup$ So to be clear, a couple has a 50% chance to have 1 kid, but if I ask a person "did your parent's have 1 kid?", they'll say yes w.p. 33.3%? Cuz if that's not a paradox, I don't know what is >.<. $\endgroup$
    – chausies
    Jan 15, 2021 at 23:46
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    $\begingroup$ Yeah it is a bit mind boggling. Another formulation that might make things more clear is if a couple had 50% chance of having no kids and 50% chance of having 1 kid. Then if you asked a kid "did your parents have 1 kid?" 100% would say yes! $\endgroup$
    – Pedro Amaral
    Jan 15, 2021 at 23:50
  • $\begingroup$ @PedroAmaral Perfect example. +1 $\endgroup$
    – user700480
    Jan 15, 2021 at 23:52
  • $\begingroup$ What are those $\;\pi_k\;$ things...? $\endgroup$
    – DonAntonio
    Jan 15, 2021 at 23:58

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