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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 threeperson 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.
 Fall '06
 MartinBazant

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