Question

In the 2020 presidential election, there are N = 1; 000; 000 eligible voters in Yolo county. Each votes with probability p = 49% for Donald Trump and with probability q = 51% for Joe Biden. Votes of different voters are independent. Assume that all voters do vote. What is the probability that Joe Biden wins Yolo county? Wining the Yolo county means getting more than half of the votes.

Answer 1

100% probability that Joe Biden wins Yolo county

Explanation:

Answer 2

100% probability that Joe Biden wins Yolo county

Explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

For a proportion q in a sample of size n, the mean is
`\mu = q` and the standard deviation is
`\sigma = (q(1-q))/(√(n))`

In this problem, we have that:

`q = 0.51, n = 1000000`

So

`\sigma = \sqrt{(0.51*0.49)/(√(1000000)) = 0.0005`

What is the probability that Joe Biden wins Yolo county?

This is the probability that he gets more than 50% = 0.5 of the votes, so it is 1 subtracted by the pvalue of Z when X = 0.5. So

`Z = (X - \mu)/(\sigma)`

`Z = (0.5 - 0.51)/(0.0005)`

`Z = -20`

`Z = -20` has a pvalue of 0

1 - 0 = 1

100% probability that Joe Biden wins Yolo county