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A Coin Game II (Posted on 2012-10-09) Difficulty: 3 of 5
Alex flips a fair coin 20 times. Bert spins a fair coin 20 + n times, with n ≥ 1. Bert wins if he gets more heads than Alex, else Alex wins. Note that Alex wins if there is a tie. What is the probability (in terms of n) that Bert wins?

No Solution Yet Submitted by K Sengupta    
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Solution A solution--could it be simplified? Comment 1 of 1

The tables below represent the following formula:

Sigma{a=0 to 20}Sigma{b=a+1 to 20+n}C(20,a)*C(20+n,b)/2^(40+n)

I haven't tried simplifying it into other than Sigma notation.

   10   for N=0 to 15
   15     Pawin=0:Pbwin=0
   20     for A=0 to 20
   25       Pa=combi(20,A)//2^20
   30       for B=0 to 20+N
   35         Pb=combi(20+N,B)//2^(20+N)
   40         if B>A then Pbwin=Pbwin+Pa*Pb:else Pawin=Pawin+Pa*Pb
   45       next B
   50     next A
   55     print N,Pbwin,Pbwin+Pawin
   60   next N

finds
 n        prob(bert wins)                   sum of probs (Alex+Bert)
 0       240416274739/549755813888              1
 1       1/2                                    1
 2       617038048193/1099511627776             1
 3       1362523998241/2199023255552            1
 4       2965189726037/4398046511104            1
 5       12741278579183/17592186044416          1
 6       27067492573429/35184372088832          1
 7       56939101650431/70368744177664          1
 8       118759443140415/140737488355328        1
 9       122943363002889/140737488355328        1
 10      63239026960593/70368744177664          1
 11      258847259372867/281474976710656        1
 12      527386413199129/562949953421312        1
 13      8564177239080199/9007199254740992      1
 14      17330709486596433/18014398509481984    1
 15      34982806339532451/36028797018963968    1
 


The last column was just a check that both added to 1.

In decimal notation:

0       0.4373146561902103712   1
1       0.5                     1
2       0.5611928356238422566   1
3       0.619604178719328047    1
4       0.6742060863955430249   1
5       0.7242578350987400881   1
6       0.7693044089316174449   1
7       0.8091533011683935683   1
8       0.8438365962633653794   1
9       0.8735651349161983603   1
10      0.8986806244677296717   1
11      0.9196101990940057646   1
12      0.9368264620930393249   1
13      0.9508146757797540926   1
14      0.9620476352554492849   1
15      0.970967926603795467    1

The following simulation verifies this for n=5:

10   for Trial=1 to 1000000
20      A=0
30      for I=1 to 20:A=A+int(rnd*2):next I
40      B=0
50      for I=1 to 25:B=B+int(rnd*2):next
60      if B>A then inc Bwin
70   next
80   print Bwin/1000000

finding

0.725066

for its million trials.

Modified for n=10 it finds 0.898571.


  Posted by Charlie on 2012-10-09 13:55:39
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