Basic Probability

Approximately 1/3 of all human twins are identical (one-egg) and 2/3 are fraternal (two-egg) twins. Identical twins are necessarily the same sex, with male and female being equally likely. Among fraternal twins, approximately one-fourth are both female, one-fourth are both male, and half are one male and one female. Finally, among all U.S. births, approximately 1 in 90 is a twin birth. Define the following events:

$A = \left\{\textrm{a U.S. birth results in twin females}\right\}$
$B = \left\{\textrm{a U.S. birth results in identical twins}\right\}$
$C = \left\{\textrm{a U.S. birth results in twins}\right\}$

(a) State in words, the event $A\cap{B}\cap{C}$.

• A U.S. birth results in identical twin females.

(b) Find $P\,\,\left(\!A\cap{B}\cap{C}\,\right)$.

• $P\,\,\left(\!A\cap{B}\cap{C}\,\right) = \frac{1}{90}\cdot \frac{1}{3}\cdot \frac{1}{2}=\frac{1}{540}$