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these. To obtain the probability, we just need to divide by the total number of paths, N (p − q, p + q ),
to obtain (p − q )/(p + q ). M. Z. Bazant – 18.366 Random Walks and Diﬀusion – Problem Set 4 Solutions
p
1
2
2
2
3
3
3 q
0
0
0
1
0
0
0 r
0
0
1
1
0
1
2 P
1.0000
1.0000
0.3333
0.1667
1.0000
0.5000
0.2000 p
3
3
3
4
4
4
4 q
1
1
2
0
0
0
0 r
1
2
2
0
1
2
3 P
0.3000
0.1333
0.0762
1.0000
0.6000
0.3333
0.1428 p
4
4
4
4
4
4
5 q
1
1
1
2
2
3
0 r
1
2
3
2
3
3
0 P
0.4000
0.2381
0.1071
0.1571
0.0762
0.0457
1.0000 p
5
5
5
5
5
5
5 q
0
0
0
0
1
1
1 r
1
2
3
4
1
2
3 P
0.6667
0.4285
0.2500
0.1111
0.4761
0.3214
0.1944 p
5
5
5
5
5
5
5 7 q
1
2
2
2
3
3
4 r
4
2
3
4
3
4
4 P
0.0889
0.2302
0.1461
0.0693
0.1004
0.0508
0.0313 Table 1: Computed probabilities for the threeperson ballot problem for p ≤ 5. q
q
q
q
q
q =0
=1
=2
=3
=4
=5 r=0
1.0000
0.7142
0.5000
0.3333
0.2000
0.0909 r=1
0.7142
0.5357
0.3890
0.2667
0.1636
0.0758 r=2
0.5000
0.3890
0.2937
0.2082
0.1313
0.0622 r=3
0.3333
0.2667
0.2082
0.1543
0.1013
0.0495 r=4
0.2000
0.1636
0.1313
0.1013
0.0712
0.0369 r=5
0.0909
0.0758
0.0622
0.0495
0.0369
0.0231 Table 2: Computed probabilities for the threeperson ballot problem for p = 6. 4.1 Simulating three candidates Appendix C simulates the threeperson voting process. All possible combinations of p, q , and r
votes less than or equal to 12 were tested, each with N = 109 trials. For an underlying process
with probability l of success, and N trials, we know that the observed number of successes will be
a binomial distribution with mean N l and variance N l(1√ l) < N/4. Thus the standard deviation
−
�
of our probability estimate is less than ( N/4)/N = 1/ 4N ≈ 1.58 × 10−5 . Thus we expect our
probabilities to be correct to four decimal places. Tables 2, 3, 4, 5, and 6 show the computed
probabilities for p = 6, p = 7, p = 8, p = 9, and p = 10 respectively, and table 1 shows the
probabilities for p ≤ 5. 4.2 A...
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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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