Twins make up a small fraction of the human population. Identical (monozygotic)

Question

Twins make up a small fraction of the human population. Identical (monozygotic) twins make up 8% of all twins.
Identical twins always have the same sex. The other 92% of twins are called dizygotic or fraternal. In a family
having two children who are not identical twins, the sex of each child is independent and each sex has probability
1=2.
a) In the cases in which the two children are not twins, what is the probability that both children are girls?
b) In the cases in which the two children are twins, what is the probability that both children are girls?
c) If the twins are both girls, what is the probability that they are identical twins?

Explanation / Answer

Let A be the event of a child being a girl and B be the event of being identical twins.

a) P(A) = 1/2

The probability of both children being girls

= 1/2 * 1/2

= 1/4

= 0.25.

b) If the two children are identical twins, they are both boys are both girls and the probability that both are girls P(A|B) = 1/2.

If the two children are not identical twins, the probability that both are girls is as normal P(A|B') = 1/4.

P(A B) = P(B) P(A|B) = 0.08*1/2

P(A B') = P(B') P(A|B') = 0.92*1/4

=> Probability that both children are girls P(A) = 0.08 * 1/2 + 0.92 * 1/4

= 0.27.

c) Probability that they are identical twins given they are both girls

P(B|A) = P(A B) / P(A)

= 0.08 * 1/2 / 0.27

= 0.1481.

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