Probability Theory: Imagine there is an election with 5 candidates and 20 voters.

J_Wood

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Hello everyone!
Imagine there is a vote. 5 candidates and 20 voters. I want to know the % probability of 2 candidates getting the same amount of votes starting from 5 votes on both candidates and so forth to 10.
Thank you!
 
Hello everyone!
Imagine there is a vote. 5 candidates and 20 voters. I want to know the % probability of 2 candidates getting the same amount of votes starting from 5 votes on both candidates and so forth to 10.
Thank you!
What are your thoughts?

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Hello everyone!
Imagine there is a vote. 5 candidates and 20 voters. I want to know the % probability of 2 candidates getting the same amount of votes starting from 5 votes on both candidates and so forth to 10.
Thank you!

The first thing we need to do is to clarify what you mean.

Are we assuming that voters vote randomly?

What does it mean "starting from 5 votes ..."? It sounds like you want separate answers for the probability that at least two of the 5 get 5 votes each, then 6 votes each, and so on, rather than what it initially sounded like, a single answer giving the probability that at least two of them get the same number of votes, regardless of what that number is.

Furthermore, am I right about "at least two", or do you want exactly two (so the others must have different numbers of votes)? Does it count if three get the same number of votes, or even all five?

Finally, if this is not for a class, can you tell us the context of the question? That could help us understand how to best interpret it for that context. In particular, would you want a method that applies only to your particular numbers, or one that can be generalized?

Truly finally, it really helps if you can tell us your background, to give us an idea of what level of explanation is needed.
 
The first thing we need to do is to clarify what you mean.

Are we assuming that voters vote randomly?

What does it mean "starting from 5 votes ..."? It sounds like you want separate answers for the probability that at least two of the 5 get 5 votes each, then 6 votes each, and so on, rather than what it initially sounded like, a single answer giving the probability that at least two of them get the same number of votes, regardless of what that number is.

Furthermore, am I right about "at least two", or do you want exactly two (so the others must have different numbers of votes)? Does it count if three get the same number of votes, or even all five?

Finally, if this is not for a class, can you tell us the context of the question? That could help us understand how to best interpret it for that context. In particular, would you want a method that applies only to your particular numbers, or one that can be generalized?

Truly finally, it really helps if you can tell us your background, to give us an idea of what level of explanation is needed.

OK, so voters vote randomly and everyone votes! And yes, I am interested in answers starting from 5 to 10 on each of 2 random candidates because that is the least number two must have in order to win the other candidates in a tie.
Just give me the probability for two candidates getting 5 votes, then 6 votes, 7,8,9,10. That's all I ask
 
OK, so voters vote randomly and everyone votes! And yes, I am interested in answers starting from 5 to 10 on each of 2 random candidates because that is the least number two must have in order to win the other candidates in a tie.
Just give me the probability for two candidates getting 5 votes, then 6 votes, 7,8,9,10. That's all I ask

You haven't answered all my questions.

Most important: It appears that what you are really asking for is the probability of a tie (with each possible number of votes). But should the probability of a tie at 5 votes each include the possibility that four candidates all get 5, or that three get 5 and the other two get 2 and 3, or do you only want two-way ties? And do you want only winning ties, as you imply here, and not, say, two getting 5 and 5, while the other three get 1, 3, and 6?

We can at least take the easiest case: two get 10 votes each. This can happen in P(5,2)*C(20,10) = 20*184,756 = 3,695,120 ways (since we can choose which two win, and then which ten vote for the first in the list), out of 5^20 = 95,367,431,640,625 possible outcomes, for a probability of 0.000000038746 = 0.0000038746%, if I've got that right.

I expect the others to be considerably harder; they will not be a matter of just plugging something into a simple formula, whichever way you define the problem.
 
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