Math Is Fun Forum

deep_blue

Member

Registered: 2009-05-17

Posts: 3

Elections Winning Odds problem

I have a probability problem that puzzles me:

There are elections for a city council, total of 50 candidates and 1000 voters. There are 8 seats available in the city council and each voter has 8 votes so that a voter must select 8 different names from 50 candidates with any order. To win the election a candidate must get votes of 40% of voters.

Candidates = 50
Voters = 1000
seats = 8
Vmin = Votes required to win = 0.4*1000 = 400

Assume that polls results were taken and that candidates were ranked according to this poll into the following ranks:
Rank1: 8 candidates  , score = 3
Rank2: 12 candidates , score = 2
Rank3: 30 candidates , score = 1

Rank1 has 3 times the chances of Rank3 and 1.5 the chances of Rank2 .. and so on.

What is the probability of a candidate from Rank1 to win the elections?
What is the probability of a candidate from Rank2 to win the elections?
What is the probability of a candidate from Rank3 to win the elections?

I tried solving the problem but didn't get convincing results, I will not show the procedure that I used to avoid affecting any received answers.

Last edited by deep_blue (2009-05-17 10:25:22)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Elections Winning Odds problem

Can I assume that Rank2 has twice the chances of Rank3?

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

deep_blue

Member

Registered: 2009-05-17

Posts: 3

Re: Elections Winning Odds problem

Yes,

There was and error in the oppenning post where I said Rank1 has twice Rank2, I have edited it.

The ratios are taken directly from score so:

R1:R2 = 3:2
R1:R3 = 3:1
R2:R3 = 2:1

Last edited by deep_blue (2009-05-17 10:28:45)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Elections Winning Odds problem

Thanks Deep;

Because the other set was inconsistent.

There are elections for a city council, total of 50 candidates and 1000 voters. There are 8 seats available in the city council and each voter has 8 votes so that a voter must select 8 different names from 50 candidates with any order. To win the election a candidate must get votes of 40% of voters.

Candidates = 50
Voters = 1000
seats = 8
Vmin = Votes required to win = 0.4*1000 = 400

Can't see how to tie this in with the rest of the problem.

The polls give us the relative probabilities of victory for the 3 groups..

P(someone from RANK1 wins) = 1/2
P(someone from RANK2 wins) = 1/3
P(someone from RANK3 wins) = 1/6

Last edited by bobbym (2009-05-17 12:45:51)

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

deep_blue

Member

Registered: 2009-05-17

Posts: 3

Re: Elections Winning Odds problem

If we collected all the votes and picked one random vote then the probability that this vote goes to a specific candidate:

P1 (one Rank1Candidate) = 3 / (3*8+2*12+1*30)
P2 (one Rank2 Candidate) = 2 / (3*8+2*12+1*30)
P3 (one Rank3 Candidate) = 1 / (3*8+2*12+1*30)

Thus all candidates in the same rank has equal chances.

Since each voter has 8 votes then the probability of RankX candidate to be voted for:

P_vote = 1 - P_novote

The probability of candidate from RankX not being voted for from 1 voter is:

P_novote = (1 - PX)^8 bcasue there are 8 votes total and not being voted for can be described as:

[(1-Px) for vote1] And [(1-Px) for Vote2] And....... And [(1-Px) for vote8)   =  (1-Px)^8

P_vote = 1 - (1-Px)^8

For each candidate in ranks group:

P1_Vote = 1 - (1 - P1)^8
P2_Vote = 1 - (1 - P2)^8
P3_Vote = 1 - (1 - P3)^8

Last edited by deep_blue (2009-05-18 03:54:00)