Question

In Pennsylvania, 6,165,478 people voted in the election. Trump received 48.18% of the vote and Clinton recieved 47.46%. This doesn't add up to 100% because other candidates received votes. All together these other candidates received 100% - 48.18% - 47.46% = 4.36% of the vote.

Suppose we could select one person at random from the 6+ million voters in PA (note: PA is the common abbreviation for Pennsylvania). We are interested in the chance that we'd choose a Trump, Clinton, or Other voter.

Below is a probability table for the choice:

Voted for	Trump	Clinton	Other
Probability	0.4818	0.4746	0.0436
Number of people	2,970,733	2,926,441	268,304

Suppose we take a simple random sample of ?=1500 n = 1500 voters from the 6+ million voters in PA. What is the expected number of Trump voters? What is the expected number of Clinton voters? To answer these questions, let ?1 T 1 be 1 if the first voter chosen for the sample voted for Trump and 0 if they voted for Clinton or another candidate. Let ?2 T 2 be 1 if the second voter chosen for the sample voted for Trump and 0 if they voted for Clinton or another candidate, and so on. Let's start with some very basic questions. Find: ?(?1000=1) P ( T 1000 = 1 ) ?(?1000=0) P ( T 1000 = 0 ) ?(?17) E ( T 17 )

Answer 1

= Probability that 1000th voter votes for Trump = 0.4818

= Probability that 1000th voter votes for Clinton or another candidate = 0.4746 + 0.0436 = 0.5182

= 1 * 0.4818 + 0 * 0.5182 = 0.4818

In general,

= 1 * 0.4818 + 0 * 0.5182 = 0.4818

Expected number of Trump voters = = 1500 * 0.4818 = 722.7

Now, C1 be 1 if the first voter chosen for the sample voted for Clinton and 0 if they voted for Trump or another candidate.

= Probability that i-th voter votes for Clinton = 0.4746

= Probability that i-th voter votes for Trump or another candidate = 0.4818 + 0.0436 = 0.5254

In general,

= 1 * 0.4746 + 0 * 0.5254 = 0.4746

Expected number of Clinton voters = = 1500 * 0.4746 = 711.9