Q: At a polling booth, ballots are cast by ten voters, of whom three are Republi

Question

Q: 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.  

The solution provided is:

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.

I'm not understanding where 5/9 for case 2 and other numbers came from? i.e. the 0,1, or 2 that is multiplied in each case.

I basically need someone to explain this solution for me!?

Explanation / Answer

I'm not understanding where 5/9 for case 2 and other numbers came from? i.e. the 0,1, or 2 that is multiplied in each case.

If first is republican, the possibilities for selection of second person are Republican, Democrat or independent.

Since 1 republican is already selected, probability of selecting republican in 2nd = 2/9 [ 9 total] . Difference = 2

Probability of democrat = 2/9 Difference =0

Probabiltiy of independted = 5/9 Difference = 1

THese 3 cases are then multiplied with their respective probabilities.

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