Equality of opportunity is related to equalized odds, which requires that both the true positive rates and false positive rates are the same for all groups.

Suppose Glubbdubdrib University admits both Lilliputians and Brobdingnagians to a rigorous mathematics program. Lilliputians' secondary schools offer a robust curriculum of math classes, and the vast majority of students are qualified for the university program. Brobdingnagians' secondary schools don't offer math classes at all, and as a result, far fewer of their students are qualified. Equality of opportunity is satisfied for the preferred label of "admitted" with respect to nationality (Lilliputian or Brobdingnagian) if qualified students are equally likely to be admitted irrespective of whether they're a Lilliputian or a Brobdingnagian.

For example, suppose 100 Lilliputians and 100 Brobdingnagians apply to Glubbdubdrib University, and admissions decisions are made as follows:

Table 1. Lilliputian applicants (90% are qualified)

Table 2. Brobdingnagian applicants (10% are qualified):

The preceding examples satisfy equality of opportunity for acceptance of qualified students because qualified Lilliputians and Brobdingnagians both have a 50% chance of being admitted.

While equality of opportunity is satisfied, the following two fairness metrics are not satisfied:

demographic parity: Lilliputians and Brobdingnagians are admitted to the university at different rates; 48% of Lilliputians students are admitted, but only 14% of Brobdingnagian students are admitted.

equalized odds: While qualified Lilliputian and Brobdingnagian students both have the same chance of being admitted, the additional constraint that unqualified Lilliputians and Brobdingnagians both have the same chance of being rejected is not satisfied. Unqualified Lilliputians have a 70% rejection rate, whereas unqualified Brobdingnagians have a 90% rejection rate.

See Fairness: Equality of opportunity in Machine Learning Crash Course for more information.