Here you can find out whether this common expression is fake or not. I created this code for using in the solution of a probability problem proposed by Dr. Ruth Urner on one of her assignments for the Machine Learning Course at YU.

The problem consists of choosing two courses out of three in two sections. First, the student was presented with two randomly chosen courses to pick one. On the second pick, they would pick another out of the remained ones (including the one not picked in the first try).

This solution was written in Java using JUnit tests. For checking the results, execute the unit test named probabilityAllThreeCourses under the class SelectorTest. For the assert of the test, we have these three probabilities:

assertEquals(new BigDecimal("0.666"), probabilityA);
assertEquals(new BigDecimal("0.666"), probabilityB);
assertEquals(new BigDecimal("0.666"), probabilityC);

Notice that all three are identical to 2/3 when we have a large number of executions, defined by the variable EXECUTIONS. The class Selector is responsible for randomly selecting our courses.
The test was made with 500000 executions, and the precision is three decimal places.