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#1 2020-10-19 09:56:56

mrpace
Member
Registered: 2012-08-16
Posts: 87

Calculate the probability of election result from 1% of votes counted

Hi there

I was watching the NZ election live the other night and wondered how confident I could be of knowing the final result given that only 1% of the votes had been counted.

Here is a simplified example

Suppose there are two parties, Party A and Party B

For a party to win the election, they must get more than 50% of votes

1,200,000 votes have been made but only 12,000 have been counted (1%)

Of those counted, Party A has received 7800 votes (65%) and Party B has received 4200 votes (35%)

What is the probability that Party A will win the election?



I appreciate any help or suggestions as to which theorem/s could help me

Thanks

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#2 2020-10-19 21:12:54

Bob
Administrator
Registered: 2010-06-20
Posts: 8,914

Re: Calculate the probability of election result from 1% of votes counted

hi mrpace

You need to create a mathematical model to 'describe' this situation.  With only two parties the binomial distribution might work.

Probability in this 'constituency' of a voter supporting A is 0.65 (p) and for B is 0.35 (q).

Now you have to assume that these probabilities will remain for the rest of the country and that's where this comes unstuck.

Expected number of votes for A =  np = 1200 000 x 0.65 = 780 000

Standard deviation is √ (npq) = 522.

You can use the normal distribution approximation for binomial results, which I have tried.  I don't think it's worth showing the calculation because it makes it virtually certain that A will win overall.

The problem is with the underlined assumption.  Think about it.  0.65 is pretty high and if that's the probability in every constituency then A will keep winning.  In the UK that doesn't happen.  There are big regional variations between support for the parties so the 'pundits' know they cannot assume general results from a single announcement. The pre-election opinion polls carefully select their samples by spreading across areas with so many people of each gender and age group.

To do what you want, you'd have to know the probabilities for each constituency; then you could calculate on an area by area basis.

The underlying theory is here: https://www.mathsisfun.com/data/standar … ution.html

The method works ok for bags of sugar if it's ok to assume consistency during the production run.  Voters are more complicated.

In the UK they do an 'exit poll' asking a sample of people how they voted after they have voted.  Providing the sample is across all areas like the opinion polls this does usually give a good prediction of the final outcome.

Bob


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