Expected number of votes

Question

There's an election where a machine said that party X received 600 votes, and party Y received 400 votes. We know the machine is 65% accurate. What is the expected number of votes that party X received?

Answer 1

We have $1000$ people let $M_i$ be the indicator function for $X$ to receive a vote from $i$-th voter and let $M$ be the number of votes party $X$ receive. Then $$M = M_1+...+M_{1000}$$ so $$E(M) = 600\cdot 0,65+ 400\cdot 0,35 =530$$

Answer 2

So 65% of the 600 votes recorded for X, 390, were actually for X and the other 210 were for Y. And 65% of the 400 votes recorded for Y, 260, were actually for Y and the other 140 were for X. That is, we would expect 390+ 140= 530 votes for X.

(That is one horribly bad voting machine!)

Answer 3

Let $X'$ be the number of votes reportedly for $X$ that were right. Let $Y'$ be the number of votes reportedly for $Y$ that were wrong. Then by linearity of expectation $$E[X'+Y']=E[X']+E[Y'].$$ Since both of these are binomial (was the vote counted correctly or not), then $E[X']=600\cdot 0.65=390$ and $E[Y']=400\cdot 0.35=140$ since $\mu=n\cdot p$. There $E[X'+Y']=530$.