Consider a county in which 100,000 people vote in an election. There are only two candidat...

Consider a county in which 100,000 people vote in an election. There are only two candidates on the ballot: a Democratic candidate (denoted D) and a Republican candidate (denoted R). As it happens, this county is heavily Democratic, so 80,000 people go to the polls with the intention of voting for D, and 20,000 go to the polls with the intention of voting for R.

However, the layout of the ballot is a little confusing, so each voter, independently and with probability , votes for the wrong candidate– that is, the one that he or she didn't intend to vote for. (Remember that in this election, there are only two candidates on the ballot.)

Let X denote the random variable equal to the number of votes received by the Democratic candidate D, when the voting is conducted with this process of error. Determine the expected value of X, and give an explanation of your derivation of this value.