Election Probability

[mmbb10010]

The question is this: There is a vote occurring for a new government position. There are only two candidates. If there are 130 voters, what is the probability of a tie is there if no other factor taken into account?
Could the person that answers also provide a formula or steps to apply this to any number of voters.

 

[pka]

The question is this: There is a vote occurring for a new government position. There are only two candidates. If there are 130 voters, what is the probability of a tie is there if no other factor taken into account?
Could the person that answers also provide a formula or steps to apply this to any number of voters.

You must know that this is not a homework service. You need to do the work.
There are \(130\) possible different vote totals. Is one of them a tie?
Assuming that the votes are all independent and equally likely, then what do you think the answer is?

 

[Dr.Peterson]

Are you allowed to assume that everyone votes? Is it assumed that each vote is random? (I wouldn't want to live in such a country.)

How many possible outcomes are there? (The answer is not 130, by the way.)

How many are ties?

What is the probability? Your teacher will have provided a formula for this ...

 

[mmbb10010]

Are you allowed to assume that everyone votes? Is it assumed that each vote is random? (I wouldn't want to live in such a country.)

How many possible outcomes are there? (The answer is not 130, by the way.)

How many are ties?

What is the probability? Your teacher will have provided a formula for this ...

This is not a homework related question. I am attempting to deal with a situation in which we believe an administrative position is facing some election fraud and I am not sure how to solve this

 

[mmbb10010]

You must know that this is not a homework service. You need to do the work.
There are \(130\) possible different vote totals. Is one of them a tie?
Assuming that the votes are all independent and equally likely, then what do you think the answer is?

This is not a homework related question it is a real situation that I guess I posed as question in which I am trying to determine the likelihood of a tie as a way to question the results of the elections in an administrative position. I found something about pivot probability online but it is too complex for me to understand as my field does not involve anything beyond basic math

 

[pka]

This is not a homework related question it is a real situation that I guess I posed as question in which I am trying to determine the likelihood of a tie as a way to question the results of the elections in an administrative position. I found something about pivot probability online but it is too complex for me to understand as my field does not involve anything beyond basic math

There is one way in \(130\) that a tie will result.
If there were \( 131 \) votes cast the the probability of a tie is zero.

 

[Dr.Peterson]

This is not a homework related question. I am attempting to deal with a situation in which we believe an administrative position is facing some election fraud and I am not sure how to solve this

What you're looking for is a hypothesis test: Is there sufficient reason to believe that the tie did not result from chance?

But since this is real life, the questions I posed that would be quietly assumed in a homework question must be dealt with: "Are you allowed to assume that everyone votes? Is it assumed that each vote is random?" (And is each vote unbiased?)

The reality will be that votes are not coin flips; but suppose they are. If we model this as 130 independent coin flips, the result is a binomial distribution. Of the 131 possible outcomes if everyone votes (from 0 for A through 130 for A, and vice versa for B), the more lopsided outcomes are less likely.

The answer, under this assumption, is P(X = 65) in a binomial distribution with p=0.5 and n=130. This is [MATH]\binom{130}{65}(0.5)^{65}(0.5)^{65} = 0.0698[/MATH]. So there is a 7% chance of this happening based on coin flips, which a statistician will generally say is reasonable to have happened by chance.

Now, if you knew that everyone had some bias in favor of A, say, then the probability would be much less, but I don't know how you'd prove that.

 

[mmbb10010]

What you're looking for is a hypothesis test: Is there sufficient reason to believe that the tie did not result from chance?

But since this is real life, the questions I posed that would be quietly assumed in a homework question must be dealt with: "Are you allowed to assume that everyone votes? Is it assumed that each vote is random?" (And is each vote unbiased?)

The reality will be that votes are not coin flips; but suppose they are. If we model this as 130 independent coin flips, the result is a binomial distribution. Of the 131 possible outcomes if everyone votes (from 0 for A through 130 for A, and vice versa for B), the more lopsided outcomes are less likely.

The answer, under this assumption, is P(X = 65) in a binomial distribution with p=0.5 and n=130. This is [MATH]\binom{130}{65}(0.5)^{65}(0.5)^{65} = 0.0698[/MATH]. So there is a 7% chance of this happening based on coin flips, which a statistician will generally say is reasonable to have happened by chance.

Now, if you knew that everyone had some bias in favor of A, say, then the probability would be much less, but I don't know how you'd prove that.

I tried to apply this to 150 and it wasn’t even close

 

[Dr.Peterson]

I can't help without seeing what you did and what you got.

But you could use Excel, or something like this: https://stattrek.com/online-calculator/binomial.aspx
For our problem, enter 0.5, 130, 65.