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There is a statistics bonus problem I had on an exam, and I got it wrong. It is as follows:

There is a county in which 100, 000 people vote in an election. There are only two candidates on the ballot: A and B. In this county, 70, 000 people go to the polls with the intention of voting for A, and the remaining 30, 000 go to the polls with the intention of voting for B. However, the layout of the ballot is a little confusing, so each voter, with probability 1/100, votes for the candidate that they did not intend to vote for. (These mistakes may be correlated: they are not necessarily independent.) Let X be the random variable equal to the number of votes received by candidate A, when the voting is conducted with such a process of error. Determine the expected value of X with a complete proof.

How do I approach a problem like this? What I did was simply subtract the number that would vote incorrectly from group A and add the number that would vote incorrectly from group B to A, but this was wrong. How does the correlation and not necessarily being independent factor into solving the question?

whuber
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John Yils
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  • Are you sure the wording is correct? Because if *every* person has a probability of 1/100 of making a mistake, then it doesn't seem like these mistakes are correlated. They are all drawn independently with probability 1/100... unless I am misunderstanding something – user3494047 Sep 23 '16 at 17:38
  • @user3494047 Simply knowing the probabilities does not imply the results are uncorrelated. For instance, it could be that in one out of every hundred elections (on average), all voters vote for the wrong person but otherwise they vote as they intended. That would be a strong correlation indeed, but it's consistent with the information given. – whuber Sep 23 '16 at 17:40
  • I see what you are saying. I guess I thought the wording implied that every voter in this election will, with a probability 1/100, make a mistake. – user3494047 Sep 23 '16 at 17:43
  • Because you seem inadvertently to have deleted the statement of the problem itself, I have rolled your post back to its original version. @user3494047 That indeed is the case in the scenario I outlined: each voter will make a mistake with probability 1/100--but that does not imply the mistakes occur independently. – whuber Sep 23 '16 at 19:26

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Let $X_i'$ be the random variable "intention of voting" (it is 1 if intention is A, 0 if B) and $Z_i$ be an indicator variable which is 0 if person votes as intended and 1 if he makes an voting error. The subscript $i$ indexes voters. Then the person $i$'s vote is $X_i= X_i' (1-Z_i) + (1-X_i') Z_i$ and $$ X = \sum_i X_i $$ and by linearity of expectation and independence of $X_i'$ and $Z_i$ (this last assumption really comes from the vote error probability not depending on the vote intent) we get $$ \DeclareMathOperator{\E}{\mathbb{E}} \E X = \sum_i \E X_i = \sum_i 0.7 \cdot 0.99 + 0.3 \cdot 0.01 \\ = 100000 (0.7 \cdot 0.99 + 0.3 \cdot 0.01) $$ and you can take it from there.

kjetil b halvorsen
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    By the way, this gives the same answer that @John Yils said they got. So their only mistake was their lack of a proof? – user3494047 Sep 23 '16 at 17:46