(a) In an election with two candidates using paper ballots,...

(a) In an election with two candidates using paper ballots, each vote is independently misrecorded with probability $p=0.02$. Use a Chernoff bound to bound the probability that more than $4 \%$ of the votes are misrecorded in an election of $1,000,000$ ballots. (b) Assume that a misrecorded ballot always counts as a vote for the other candidate. Suppose that candidate A received 510,000 votes and that candidate B received 490,000 votes. Use Chernoff bounds to bound the probability that candidate B wins the election owing to misrecorded ballots. Specifically, let $X$ be the number of votes for candidate $\mathrm{A}$ that are misrecorded and let $Y$ be the number of votes for candidate $\mathrm{B}$ that are misrecorded. Bound $\operatorname{Pr}((X>k) \cap(Y<\ell))$ for suitable choices of $k$ and $\ell$.

(a) In an election with two candidates using paper ballots, each vote is independently misrecorded with probability $p=0.02$. Use a Chernoff bound to bound the probability that more than $4 \%$ of the votes are misrecorded in an election of $1,000,000$ ballots.
(b) Assume that a misrecorded ballot always counts as a vote for the other candidate. Suppose that candidate A received 510,000 votes and that candidate B received 490,000 votes. Use Chernoff bounds to bound the probability that candidate B wins the election owing to misrecorded ballots. Specifically, let $X$ be the number of votes for candidate $\mathrm{A}$ that are misrecorded and let $Y$ be the number of votes for candidate $\mathrm{B}$ that are misrecorded. Bound $\operatorname{Pr}((X>k) \cap(Y<\ell))$ for suitable choices of $k$ and $\ell$.

Key Concepts

Chernoff Bound

The Chernoff bound is a method for providing exponentially decreasing bounds on the tail probabilities of sums of independent random variables. It is especially useful for estimating the probability that such a sum deviates significantly from its expected value, even when the variables are not exactly normally distributed. In the context of the election problem, the Chernoff bound is used to limit the probability that the total count of misrecorded ballots deviates far above its mean.

Binomial Distribution

The binomial distribution arises when modeling the number of successes in a fixed number of independent Bernoulli trials, where each trial has only two possible outcomes (e.g., a ballot being misrecorded or not). In the election scenario, each ballot is an independent trial with a 2% chance of error, making the total misrecorded ballots follow a binomial distribution.

Tail Bounds

Tail bounds are techniques used to estimate the probability that a random variable deviates significantly from its mean. Chernoff bounds are a prime example, providing a way to handle the extreme tails of the distribution of sums such as the total number of misrecorded ballots. They allow one to bound uncommon events, like a high number of misrecordings, without requiring exact calculations.

Independence of Random Variables

Many of the probabilistic bounds, including the Chernoff bound, depend critically on the assumption that the random variables in question are independent. In the election context, each ballot is assumed to have been processed independently, ensuring that the event of one ballot being misrecorded does not affect another. This independence is crucial for applying tail bounds to the sum of misrecorded ballots.

Joint Error Events and Union Bound

When assessing the simultaneous occurrence of two events (such as one candidate’s votes being misrecorded above a threshold and another candidate’s votes being misrecorded below a threshold), it is often required to consider joint error events. Techniques like the union bound or the use of independent probability estimates allow one to bound the probability that both events happen concurrently. This is important in the second part of the problem where the joint likelihood of errors causing an upset is evaluated.

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