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Take a given chromosome that is only present in one of the parents of a child. Assume that the probability of one child inheriting this chromosome is ${1}\over{2}$. At the beginning of the pregnancy, before testing, the probability that both children inherit this chromosome is clearly ${1}\over{4}$.

Partway through the pregnancy a test is done to determine if either twin has inherited the chromosome. The test can only determine the presence of the chromosome in the bloodstream and so can only determine if at least one twin has the chromosome. It comes back positive, showing that at least one of the children has inherited the chromosome.

Given this test result, what is the probability that both twins have inherited the chromosome?

Sean Seefried
  • 103
  • Similar to the classic https://math.stackexchange.com/questions/15055/in-a-family-with-two-children-what-are-the-chances-if-one-of-the-children-is-a though here perhaps translatable as boys and the Y chromosome – Henry Jun 01 '20 at 09:34
  • @Henry Indeed. This precise scenario actually happened to me. We knew that at least one of our twins was a boy. In the end it turned out both were. Looking at the classic question you referred to, there seems to be a consensus that if the question is not sufficiently unambiguously phrased that the answer can differ. I would like to know if you think this question is sufficiently unambiguous or whether it could be improved. – Sean Seefried Jun 01 '20 at 22:24

1 Answers1

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Hint:

  • You have suggested $\frac14$ of such pregnancies have both children inheriting the chromosome.
  • What proportion of such pregnancies have neither inheriting the chromosome?
  • What proportion of such pregnancies have at least one inheriting the chromosome?
  • What proportion of pregnancies with at least one inheriting the chromosome have both children inheriting the chromosome?
Henry
  • 157,058
  • Thanks for the hints. I think the answer is 1/3 but I have colleagues that insist it must be 1/2. This is not a homework question. (I am 42 and work full-time.) If it's not too much to ask I'd really like to see a worked answer. – Sean Seefried Jun 01 '20 at 00:32
  • I'd also be interested in determining whether the question, as it is currently phrased, is ambiguous or not. – Sean Seefried Jun 01 '20 at 00:33
  • The answer is indeed $\frac13$ as being $\frac{1/4}{3/4}$ and is a basic application of conditional probability. This uses your assumption of $\frac12$ for each child inheriting the chromosome and the additional assumption of independence between the children which is implied by your $\frac14$ for both children inheriting the chromosome. – Henry Jun 01 '20 at 07:41