Start with a bag containing 5 white beans. Randomly draw one at a time employing the following rule:
If the bean is white, color it black and put it back in the bag;
If the bean is black, keep it out.
What is the probability that at some point there will be a single white bean in the bag?
Generalize to start with N beans.
Does the probability converge, and if so, to what value?
(In reply to
computer aided solution by Charlie)
If there are n beans to start with and b beans are left after d draws, then, since n-b beans have disappeared, 2*(n-b) of the d draws have been used to disappear beans, the number of black beans left is d - 2*(n-b) and the number of white beans is then b - d + 2*(n-b) = 2*n - b - d
This allows us to use a 1-dimensional array of number of beans remaining, and still know how many are white and how many are black and any stage, d, of the procedure. That in turn allows use of UBASIC for larger values of n, with its greater precision and its storage of exact rational numbers, despite its limitation of total elements to 200. It reports the probability that only 1 bean remains after 2*(n-1) draws, rather than 2, as one bean would be white, but 2 beans would be black.
In the formula for black beads in the program, D-1 rather than D is used, as the draw being calculated has not taken place prior to this draw.
4 kill "beancolr.txt"
5 open "beancolr.txt" for output as #2
10 dim Oldb(90),Newb(90)
20 for N=2 to 30
30 erase Oldb()
40 dim Oldb(N)
50 Oldb(N)=1
60 for D=1 to 2*(N-1)
70 erase Newb()
80 dim Newb(N)
90 for B=2 to N
100 if Oldb(B)>0 then
110 :Black=D-1-2*(N-B):White=B-Black
120 :Newb(B-1)=Newb(B-1)+Oldb(B)*Black//B
130 :Newb(B)=Newb(B)+Oldb(B)*White//B
140 :endif
150 next B
160 for B=1 to N
170 Oldb(B)=Newb(B)
180 next B
190 next D
200 print N,Newb(1),Newb(1)/1
205 print #2,N,Newb(1),Newb(1)/1
210 next N
220 close #2
I've modified the output to replace double-/ with single /, and truncated the results (cutting off at n=19) so the large numbers would not greatly expand the window allotted. For the larger ones, I've separated the numerator and denominator onto two lines.
2 1/2 0.5
3 7/18 0.3888888888888888888
4 97/288 0.3368055555555555555
5 54997/180000 0.3055388888888888888
6 460463/1620000 0.2842364197530864196
7 51185279267/190591380000 0.2685603056497098661
8 200170674968477/780662292480000 0.2564113534068320931
9 11369422537341929933/46097327708651520000 0.2466395147501819988
10 6873027837417268175729/28810829817907200000000 0.2385570940114115901
11 173163272842349985846536059219169/747278726094210559615027200000000 0.2317251472518421861
12 168772133656672827075008567368139/747278726094210559615027200000000 0.2258489741020614426
13 3842815959898463365301025948437541787295761139
/17410163370762081276553474535120146483200000000 0.2207225675062838212
14 442833976487751453451845289533115618663180920930551
/2048288310406788100105239725582350113601996800000000 0.2161970920977452871
15 271605621773967287861163059327877888253955472822789551
/1280180194004242562565774828488968821001248000000000000 0.2121620245696967696
16 1146585683227688521430368531418768743587440561958751522161688661
/5498332066235157091471236737260130182964038135185408000000000000 0.2085333641939861734
17 54914670162209509582192304011109671718184598872345633534579943119292785699166225061
/267555391671200852451892607513749958318859975038049807576997165686325248000000000000
0.2052459859590279339
18 258819282870746576472902196651366139225248527085967063366380554634931856876858842095501029
/1279709144146211870050976333067433124390399375947766050096742556565557385101312000000000000
0.2022485219041112374
19 26583896812182990412714688743262439366381795231421715900029755514387245409789265482456656860178254671498968825629
/133252722331952794500479894206857954240573271518729997224070375088768946767085399118394931282960187392000000000000
0.1994998402055791471
Edited on July 14, 2015, 10:52 am
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Posted by Charlie
on 2015-07-14 10:42:50 |
Further exact values using UBASIC
