Two candidates were running for a post. The voting machines recorded 520,000 votes for the first candidate and 480,000 votes for the second one. Afterwards it became apparent that the voting machines were defective— they randomly and independently switched each vote for the opposite one with probability of 45%. The losing candidate asked for a re-vote. Is there a basis for a re-vote?
Source: Theory of probability and random processes / leonid b. koralov, yakov g. sinai. Section 2.4 Question 7
My attempt: I know p=0.45 is the parameter in the binomial distribution. But I don't know how to build such a binomial distribution. I am thinking a basis for a re-vote can be there are 2000 votes from the first candidate should belong to the second candidate. Then I am studying $P(X \geq 2000)$ where X is Binom(520000, 0.45). Is this thinking correct?
The defective voting machines can switch votes both ways -- a vote for candidate 1 can be switched to a vote for candidate 2, and vice versa. So the number of intended votes for candidate 2 equals the random variable $X+Y$, where $X$ is the number of recorded votes for candidate 1 that in fact were switched from candidate 2, and $Y$ is the number of recorded votes for candidate 2 that were not switched to candidate 1. Then $X$ and $Y$ are independent binomial variables; note they have different success probabilities.
You want the probability $P(X+Y>500,000)$ since $\{X+Y>500,000\}$ is the event that candidate 2 received more intended votes than candidate 1. You should use the normal approximation to compute this probability.