problemset4

The total number of possible voting processes

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Unformatted text preview: nalytical results for three walkers Consider the case when r = 1. The total number of possible ways the votes can be counted is (p + q + 1)N (p − q , p + q ), since any counting process can be viewed as a counting process between candidates A and B only, with C’s vote inserted at one of p + q + 1 located between the other votes. We know that in order for A to always be ahead, he must receive the first two votes. Consider any voting process between A and B where A is always ahead. If C’s vote is inserted before any votes are counted, then C will take the lead and A will not always be ahead. If C’s vote is inserted after one vote has been counted, then C will tie with A, and again the condition will be violated. However, if C’s vote is inserted at any later point, then it will not violate the condition, since A must have at least two votes by this stage. The total number of possible voting processes satisfying the condition in therefore (p + q − 1)F (p − q , p + q ), and hence the probability of the condition being satisfied is (p + q − 1)(p − q ) (p + q − 1)F (p − q , p + q ) = . (p + q + 1)N (p − q , p + q ) (p + q + 1)(p + q ) For the case when r > 1 the reader should refer to references [2] and [3]. M. Z. Bazant – 18.366 Random Walks and Diffusion – Problem Set 4 Solutions q q q q q q q =0 =1 =2 =3 =4 =5 =6 r=0 1.0000 0.7500 0.5556 0.4000 0.2727 0.1667 0.0769 r=1 0.7500 0.5835 0.4444 0.3272 0.2273 0.1411 0.0659 r=2 0.5556 0.4444 0.3484 0.2631 0.1868 0.1179 0.0559 r=3 0.4000 0.3272 0.2631 0.2047 0.1490 0.0964 0.0466 r=4 0.2727 0.2273 0.1868 0.1490 0.1128 0.0755 0.0376 r=5 0.1667 0.1411 0.1179 0.0964 0.0755 0.0539 0.0284 8 r=6 0.0769 0.0659 0.0559 0.0466 0.0376 0.0284 0.0179 Table 3: Computed probabilities for the three-person ballot problem for p = 7. q q q q q q q q =0 =1 =2 =3 =4 =5 =6 =7 r=0 1.0000 0.7778 0.6000 0.4544 0.3334 0.2309 0.1429 0.0667 r=1 0.7778 0.6222 0.4906 0.3788 0.2824 0.1979 0.1238 0.0583 r=2 0.6000 0.4906 0.3960 0.3120 0.2...
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This note was uploaded on 01/23/2014 for the course MATH 18.366 taught by Professor Martinbazant during the Fall '06 term at MIT.

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