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I'm working on a problem from a textbook (self-study) (Casella and Berger).

The Problem

Given these facts/assumptions...

  • ⅓ of human twins are identical (one-egg) and ⅔ are fraternal (two-egg).
  • identical twins are the same sex with sexes being equally likely.
  • among fraternal twins, ¼ are female, ¼ are male, and ½ are male/female.
  • among all US births, 1/90 are twins.

Let:

$$ \begin{align} A\ &=\ \{Births\ which\ Result\ in\ Twin\ Females\} \\ B\ &=\ \{Births\ which\ Result\ in\ Identical\ Twins\} \\ C\ &=\ \{Births\ which\ Result\ in\ Twins\} \end{align} $$

and find P($A \cap B \cap C$), i.e. the probability of identical twin females.

My attempt to solve

The first thing I tried to do was translate the above facts into probability statements that can be used to calculate P($A \cap B \cap C$).

  • ⅓ of human twins are identical (one-egg) and ⅔ are fraternal (two-egg).

Given a set of twins, a fraction of them are either identical or fraternal:

$$ \begin{align} P(identical\ |\ twins) &= \frac{1}{3} \\ P(fraternal\ |\ twins) &= \frac{2}{3} \\ \end{align} $$

  • identical twins are the same sex with sexes being equally likely.

$$ \begin{align} P(female\ |\ identical\ twins)\ = P(male\ |\ identical\ twins) = \frac{1}{2} \end{align} $$

  • among fraternal twins, ¼ are female, ¼ are male, and ½ are male/female.

$$ \begin{align} P(female\ |\ fraternal\ twins) = P(male\ |\ fraternal\ twins) &= \frac{1}{4} \\ P(male/female\ |\ fraternal\ twins) &= \frac{1}{2} \end{align} $$

  • among all US births, 1/90 are twins.

$$ P(twins) = \frac{1}{90} $$

Stuck on this one... do $A$, $B$ and $C$ have to be calculated individually in order to calculate P($A \cap B \cap C$)?

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    $\begingroup$ Your notation is vague: could you explain what a set of "birth results" might refer to in your definitions of $A,B,C$? Are you just trying to compute the proportion of US births that are identical twin girls? $\endgroup$
    – whuber ♦
    Feb 10, 2018 at 13:55
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    $\begingroup$ @whuber I updated the terms for the definitions of $A$, $B$ and $C$. Yes, I believe the problem is just asking for the probability of births that are identical twin girls. $\endgroup$
    – quantif
    Feb 10, 2018 at 14:07
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    $\begingroup$ This seems like it could be a homework problem, so I don't want to provide a full answer. But as a hint, try factoring the probabilities. Remember that you can express intersections in terms of conditional probabilities. For example, $P(E \cap F) = P(E | F) P(F)$. $\endgroup$
    – Kori K
    Feb 10, 2018 at 15:15

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