problemset4

0000 07142 05000 03333 02000 00909 r1 07142 05357

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Unformatted text preview: )/(p+q ) of 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 three-person 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 three-person ballot problem for p = 6. 4.1 Simulating three candidates Appendix C simulates the three-person 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.

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