All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars
 perplexus dot info

 A Coin Game II (Posted on 2012-10-09)
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 No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
 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   11       0.5                     12       0.5611928356238422566   13       0.619604178719328047    14       0.6742060863955430249   15       0.7242578350987400881   16       0.7693044089316174449   17       0.8091533011683935683   18       0.8438365962633653794   19       0.8735651349161983603   110      0.8986806244677296717   111      0.9196101990940057646   112      0.9368264620930393249   113      0.9508146757797540926   114      0.9620476352554492849   115      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

 Search: Search body:
Forums (0)