9853

Probability of Election in a Vote

A group of people wishes to elect one of their members as their leader. Each member of the group can vote for any member in a smaller set of candidates of size . Candidates can vote for themselves. At least votes are needed to elect a leader. Suppose that each of the people votes for one of the candidates randomly. The result shows the approximate probability of election as a function of .

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
  • [Snapshot]

DETAILS

Snapshot 1: There are 7 voters and 5 votes are required for election. For 2 candidates, . For 3 candidates, decreases to . For 7 candidates (that is, everyone is a candidate), then .
Snapshot 2: Again there are 7 voters, but now only a simple majority (4) is needed for election. This is the so-called imperial election problem [1]. With 2 candidates, one of them always gets at least 4 votes, so that the election is certain. For 3 candidates, , and for 7 candidates, .
Snapshot 3: There are 7 voters as in snapshot 2, but now only 3 votes are required for election. With 2 or 3 candidates, one of them always gets at least 3 votes, so that the election is certain. For 4 candidates, , and even for 7 candidates is as high as 0.438.
Snapshot 4: There are 25 voters, and 17 votes (two-thirds of the votes) are needed for election. This is the so-called pope problem [1] that refers to the papal election in 1513. For 2 and 3 candidates, the probabilities of election are approximately 0.109 and 0.0013, respectively.
The Demonstration is based on problem 19 in [1].
Reference
[1] P. J. Nahin, Digital Dice: Computational Solutions to Practical Probability Problems, Princeton: Princeton University Press, 2008.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.









 
RELATED RESOURCES
Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2014 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+