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Elvis had a twin brother (who died at birth).

Historically, approximately $1/125$ of all births were fraternal twins and 1/300 were identical twins. The probability that Elvis was an identical twin is approximately . . .

I'm trying to understand how to apply Bayes Theorem. To try to solve this problem above, I tried to apply Bayes Theorem by plugging in values.

Let $I$ be: Being an identical twin

Let $B$ be: Having an identical twin brother

We want to calculate $P(I|B)$.

Using Bayes, $P(I|B)=P(I)*P(B|I)/P(B)$

$P(I)=1/300$

$P(B|I)=1$ because we know that Elvis is male, so the chance of having an twin brother if we assume Elvis is an identical twin is $100\%$

$P(B)=?$

How do I calculate what the chance of having an twin brother is generally? It seems like I can't apply Bayes theorem if I can't calculate this value.

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  • $\begingroup$ I agree that they should have given you some information on the gender distribution of fraternal twins. After all, if his twin was female we'd know the twins were non-identical, so saying "brother" is evidence for the twins being identical. I suppose you could simply assume that fraternal twins satisfy the simplest statistics ($50\%$ MF and $25\%$ each of $MM$ and $FF$) and that identical twins are $50\%$ each of $MM$ and $FF$, though I do not believe these assumptions are strictly accurate. $\endgroup$
    – lulu
    May 19 '20 at 23:22
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There is a slight ambiguity in your data (are births completed pregnancies or children?) but it does not affect the answer. Also, more boys are born than girls and such patterns may be different with twins, but let's assume not here

So if

  • probability of a child being an identical twin (and twin sibling being the same sex) is $\frac1{300} \times 1$

  • probability of a child being a fraternal twin and twin sibling being the same sex is $\frac1{125}\times \frac12$

then the probability of a child being an identical twin given a twin sibling is the same sex is $$\frac{\frac1{300} \times 1}{\frac1{300} \times 1+\frac1{125}\times \frac12} =\frac{5}{11}$$

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  • $\begingroup$ Thanks @Henry, I understand now how this fits with Bayes Theroem. If I is probability of being an identical twin, and T is probability of a child having a twin sibling that is the same sex, then then P(I|T)=\frac{P(I)P(T|I)}{P(T)}. P(I)=1/300, P(T|I)=1, and P(T)=1/300+1/250 as you calculated above. $\endgroup$
    – leontp587
    May 19 '20 at 23:53
  • $\begingroup$ @leontp587 yes - I have edited in a $\times 1$ for your point $\endgroup$
    – Henry
    May 20 '20 at 0:10

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