At a polling booth, ballots are cast by ten voters, of whom three are Republicans, two are Democrats, and five are Independents. A local journalist interviews two of these voters, chosen randomly.

Calculate the expectation of the absolute value of the difference between the number of Republicans interviewed and the number of Democrats interviewed.

• 1/5
• 7/15
• 3/5
• 11/15
• 1

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: D

Consider three cases, one for each result of the first interview.

Independent (prob 0.5): Expected absolute difference is (4/9)(0) +(5/9)(1) = 5/9

Republican (prob =0.3): Expected absolute difference is (2/9)(0) + (5/9)(1) + (2/9)(2) = 1

Democrat (prob = 0.2): Expected absolute difference is (3/9)(0) + (5/9)(1) + (1/9)(2) = 7/9.

The unconditional expectation is 0.5(5/9) + 0.3(1) + 0.2(7/9) = 6.6/9 = 11/15.

Alternatively, the six possible outcomes can be listed along with their probabilities and absolute differences.

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.