My uncle’s ritual for dressing each morning except Sunday includes a trip to the sock drawer, where he:
(1) picks out three socks at random, then
(2) wears any matching pair and returns the odd sock to the drawer or
(3) returns the three socks to the drawer if he has no matching pair and repeats steps (1) and (3) until he completes step (2).
The drawer starts with 16 socks each Monday morning (eight blue, six black, two brown) and ends up with four socks each Saturday evening.
(a) On which day of the week does he average the longest time to dress?
(b) On which day of the week is he least likely to get a pair from the first
three socks chosen?
Source: manchi tutorials
5 kill "unclhabt.txt":open "unclhabt.txt" for output as #2
10 dim ExVal(6),ExProb(6)
20 gosub *AddLevel(1,1,8,6,2)
30 for I=1 to 6
35 print ExVal(I)
40 next:print
45 for I=1 to 6
50 print ExVal(I)/1
55 next
60 print
130 for I=1 to 6
135 print ExProb(I)
140 next:print
145 for I=1 to 6
150 print ExProb(I)/1
155 next
160 print
165 close #2
170 end
180
410 *AddLevel(Day,Probi,Blue,Black,Brown)
413 local Socks,Sock3,P2blue,P2black,P2brown,P3diff
420 Socks=Blue+Black+Brown:Sock3=Socks*(Socks-1)*(Socks-2)
430 if Blue>=3 then
440 :P3blue=Blue*(Blue-1)*(Blue-2)//Sock3
450 :else
460 :P3blue=0
470 if Black>=3 then
480 :P3black=Black*(Black-1)*(Black-2)//Sock3
490 :else
500 :P3black=0
510 if Brown>=3 then
520 :P3brown=Brown*(Brown-1)*(Brown-2)//Sock3
530 :else
540 :P3brown=0
550 if Blue>=2 and Brown>=1 then
560 :P2blue1brown=3*Blue*(Blue-1)*Brown//Sock3
570 :else
580 :P2blue1brown=0
590 if Blue>=2 and Black>=1 then
600 :P2blue1black=3*Blue*(Blue-1)*Black//Sock3
610 :else
620 :P2blue1black=0
630 if Black>=2 and Blue>=1 then
640 :P2black1blue=3*Black*(Black-1)*Blue//Sock3
650 :else
660 :P2black1blue=0
670 if Black>=2 and Brown>=1 then
680 :P2black1brown=3*Black*(Black-1)*Brown//Sock3
690 :else
700 :P2black1brown=0
710 if Brown>=2 and Blue>=1 then
720 :P2brown1blue=3*Brown*(Brown-1)*Blue//Sock3
730 :else
740 :P2brown1blue=0
750 if Brown>=2 and Black>=1 then
760 :P2brown1black=3*Brown*(Brown-1)*Black//Sock3
770 :else
780 :P2brown1black=0
790 P3diff=1-P3blue-P3black-P3brown-P2blue1black-P2blue1brown
800 P3diff=P3diff-P2black1brown-P2black1blue-P2brown1blue-P2brown1black
810 ExVal(Day)=ExVal(Day)+Probi//(1-P3diff)
815 ExProb(Day)=ExProb(Day)+Probi*(1-P3diff)
816 if Probi<>0 then print #2,Day,using(2,5),Probi/1,using(3,0),Blue,Black,Brown,using(3,5),P3diff/1,(1-P3diff)/1,1/(1-P3diff),Probi/(1-P3diff),Probi*(1-P3diff)/1
820 P2blue=(P3blue+P2blue1brown+P2blue1black)//(1-P3diff)
830 P2black=(P3black+P2black1brown+P2black1blue)//(1-P3diff)
840 P2brown=(P3brown+P2brown1blue+P2brown1black)//(1-P3diff)
850
860 if Day<6 then
870 :gosub *AddLevel(Day+1,Probi*P2blue,Blue-2,Black,Brown)
880 :gosub *AddLevel(Day+1,Probi*P2black,Blue,Black-2,Brown)
890 :gosub *AddLevel(Day+1,Probi*P2brown,Blue,Black,Brown-2)
900
910 return
finds the expected number of tries in order to get a pair of matching socks:
day expected number of tries
1 35/29
2 38962/31755
3 80293631/64176855
4 164300677/128353710
5 5853613/4477455
6 18248918/13432365
1 1.2068965517241379309
2 1.2269563848212879861
3 1.2511306607343099003
4 1.2800617683742838442
5 1.3073527260463812589
6 1.3585781803874448021
and the probability of finding a matching pair on the first try:
day probability of success on first try
1 29/35
2 2154/2639
3 56077/69861
4 759690163/962652825
5 698897/895491
6 17570722/22387275
1 0.8285714285714285714
2 0.816218264494126563
3 0.8026939207855598975
4 0.7891631783244390312
5 0.7804623385382991007
6 0.7848530917675331186
so Saturday has the longest expected number of tries needed, and presumably the longest time to dress. Friday is the day with the lowest expectation of success on the first try. This is paradoxical, but is related to the fact that the average of a set of numbers is not equal to the reciprocal of the average of their reciprocals; in other words the arithmetic mean is not the same as the harmonic mean.
Below is a breakdown of the days' probabilities, sorted from the output file. There are a few things to notice. For example, the .56288 probability that, on the third day there will initially be 6 blue, 4 black and 2 brown socks in the drawer, is broken down into two parts, with probabilities 0.27356 and 0.28932. That's because there are two ways of arriving at that state: the first day 2 blue socks were used and the second day 2 black, or the other way around. Needless to say, once this state is achieved, the probabilities are the same, but the contributions to the overall expected value are proportional to the probabilities of arriving at the given state.
day prob blue brwn that day's conditional overall overall
of blk P(fail) P(succ) contrib to contrib to contrib to
reaching exp value exp value prob of finding
1 1.00000 8 6 2 0.17143 0.82857 1.20690 1.20690 0.82857
2 0.60345 6 6 2 0.19780 0.80220 1.24658 0.75224 0.48408
2 0.36638 8 4 2 0.17582 0.82418 1.21333 0.44454 0.30196
2 0.03017 8 6 0 0.00000 1.00000 1.00000 0.03017 0.03017
3 0.28932 4 6 2 0.21818 0.78182 1.27907 0.37007 0.22620
3 0.27356 6 4 2 0.21818 0.78182 1.27907 0.34991 0.21388
3 0.28932 6 4 2 0.21818 0.78182 1.27907 0.37007 0.22620
3 0.01857 6 6 0 0.00000 1.00000 1.00000 0.01857 0.01857
3 0.02480 6 6 0 0.00000 1.00000 1.00000 0.02480 0.02480
3 0.07816 8 2 2 0.14545 0.85455 1.17021 0.09146 0.06679
3 0.01466 8 4 0 0.00000 1.00000 1.00000 0.01466 0.01466
3 0.01160 8 4 0 0.00000 1.00000 1.00000 0.01160 0.01160
4 0.08747 2 6 2 0.20000 0.80000 1.25000 0.10934 0.06998
4 0.18503 4 4 2 0.26667 0.73333 1.36364 0.25232 0.13569
4 0.17495 4 4 2 0.26667 0.73333 1.36364 0.23857 0.12830
4 0.18503 4 4 2 0.26667 0.73333 1.36364 0.25232 0.13569
4 0.01240 4 6 0 0.00000 1.00000 1.00000 0.01240 0.01240
4 0.01682 4 6 0 0.00000 1.00000 1.00000 0.01682 0.01682
4 0.00928 4 6 0 0.00000 1.00000 1.00000 0.00928 0.00928
4 0.08747 6 2 2 0.20000 0.80000 1.25000 0.10934 0.06998
4 0.08271 6 2 2 0.20000 0.80000 1.25000 0.10338 0.06616
4 0.06985 6 2 2 0.20000 0.80000 1.25000 0.08731 0.05588
4 0.01682 6 4 0 0.00000 1.00000 1.00000 0.01682 0.01682
4 0.00886 6 4 0 0.00000 1.00000 1.00000 0.00886 0.00886
4 0.00928 6 4 0 0.00000 1.00000 1.00000 0.00928 0.00928
4 0.01240 6 4 0 0.00000 1.00000 1.00000 0.01240 0.01240
4 0.01590 6 4 0 0.00000 1.00000 1.00000 0.01590 0.01590
4 0.01119 6 4 0 0.00000 1.00000 1.00000 0.01119 0.01119
4 0.00416 8 0 2 0.00000 1.00000 1.00000 0.00416 0.00416
4 0.00274 8 2 0 0.00000 1.00000 1.00000 0.00274 0.00274
4 0.00416 8 2 0 0.00000 1.00000 1.00000 0.00416 0.00416
4 0.00346 8 2 0 0.00000 1.00000 1.00000 0.00346 0.00346
5 0.00729 0 6 2 0.00000 1.00000 1.00000 0.00729 0.00729
5 0.07289 2 4 2 0.28571 0.71429 1.40000 0.10205 0.05207
5 0.08411 2 4 2 0.28571 0.71429 1.40000 0.11775 0.06008
5 0.08411 2 4 2 0.28571 0.71429 1.40000 0.11775 0.06008
5 0.07952 2 4 2 0.28571 0.71429 1.40000 0.11133 0.05680
5 0.00729 2 6 0 0.00000 1.00000 1.00000 0.00729 0.00729
5 0.00413 2 6 0 0.00000 1.00000 1.00000 0.00413 0.00413
5 0.00561 2 6 0 0.00000 1.00000 1.00000 0.00561 0.00561
5 0.00309 2 6 0 0.00000 1.00000 1.00000 0.00309 0.00309
5 0.08411 4 2 2 0.28571 0.71429 1.40000 0.11775 0.06008
5 0.07952 4 2 2 0.28571 0.71429 1.40000 0.11133 0.05680
5 0.06892 4 2 2 0.28571 0.71429 1.40000 0.09649 0.04923
5 0.05820 4 2 2 0.28571 0.71429 1.40000 0.08149 0.04157
5 0.08411 4 2 2 0.28571 0.71429 1.40000 0.11775 0.06008
5 0.07289 4 2 2 0.28571 0.71429 1.40000 0.10205 0.05207
5 0.01590 4 4 0 0.00000 1.00000 1.00000 0.01590 0.01590
5 0.00827 4 4 0 0.00000 1.00000 1.00000 0.00827 0.00827
5 0.00619 4 4 0 0.00000 1.00000 1.00000 0.00619 0.00619
5 0.00591 4 4 0 0.00000 1.00000 1.00000 0.00591 0.00591
5 0.00827 4 4 0 0.00000 1.00000 1.00000 0.00827 0.00827
5 0.01682 4 4 0 0.00000 1.00000 1.00000 0.01682 0.01682
5 0.01060 4 4 0 0.00000 1.00000 1.00000 0.01060 0.01060
5 0.00746 4 4 0 0.00000 1.00000 1.00000 0.00746 0.00746
5 0.00619 4 4 0 0.00000 1.00000 1.00000 0.00619 0.00619
5 0.01121 4 4 0 0.00000 1.00000 1.00000 0.01121 0.01121
5 0.01682 4 4 0 0.00000 1.00000 1.00000 0.01682 0.01682
5 0.01121 4 4 0 0.00000 1.00000 1.00000 0.01121 0.01121
5 0.00582 6 0 2 0.00000 1.00000 1.00000 0.00582 0.00582
5 0.00689 6 0 2 0.00000 1.00000 1.00000 0.00689 0.00689
5 0.00729 6 0 2 0.00000 1.00000 1.00000 0.00729 0.00729
5 0.00388 6 0 2 0.00000 1.00000 1.00000 0.00388 0.00388
5 0.00256 6 2 0 0.00000 1.00000 1.00000 0.00256 0.00256
5 0.00413 6 2 0 0.00000 1.00000 1.00000 0.00413 0.00413
5 0.00373 6 2 0 0.00000 1.00000 1.00000 0.00373 0.00373
5 0.00530 6 2 0 0.00000 1.00000 1.00000 0.00530 0.00530
5 0.00323 6 2 0 0.00000 1.00000 1.00000 0.00323 0.00323
5 0.00295 6 2 0 0.00000 1.00000 1.00000 0.00295 0.00295
5 0.00729 6 2 0 0.00000 1.00000 1.00000 0.00729 0.00729
5 0.00689 6 2 0 0.00000 1.00000 1.00000 0.00689 0.00689
5 0.00309 6 2 0 0.00000 1.00000 1.00000 0.00309 0.00309
5 0.00388 6 2 0 0.00000 1.00000 1.00000 0.00388 0.00388
5 0.00582 6 2 0 0.00000 1.00000 1.00000 0.00582 0.00582
5 0.00561 6 2 0 0.00000 1.00000 1.00000 0.00561 0.00561
5 0.00018 8 0 0 0.00000 1.00000 1.00000 0.00018 0.00018
5 0.00023 8 0 0 0.00000 1.00000 1.00000 0.00023 0.00023
5 0.00028 8 0 0 0.00000 1.00000 1.00000 0.00028 0.00028
5 0.00028 8 0 0 0.00000 1.00000 1.00000 0.00028 0.00028
6 0.01262 0 4 2 0.00000 1.00000 1.00000 0.01262 0.01262
6 0.01262 0 4 2 0.00000 1.00000 1.00000 0.01262 0.01262
6 0.01093 0 4 2 0.00000 1.00000 1.00000 0.01093 0.01093
6 0.00651 0 4 2 0.00000 1.00000 1.00000 0.00651 0.00651
6 0.01193 0 4 2 0.00000 1.00000 1.00000 0.01193 0.01193
6 0.00044 0 6 0 0.00000 1.00000 1.00000 0.00044 0.00044
6 0.00078 0 6 0 0.00000 1.00000 1.00000 0.00078 0.00078
6 0.00078 0 6 0 0.00000 1.00000 1.00000 0.00078 0.00078
6 0.00033 0 6 0 0.00000 1.00000 1.00000 0.00033 0.00033
6 0.00060 0 6 0 0.00000 1.00000 1.00000 0.00060 0.00060
6 0.05887 2 2 2 0.40000 0.60000 1.66667 0.09812 0.03532
6 0.05102 2 2 2 0.40000 0.60000 1.66667 0.08504 0.03061
6 0.05887 2 2 2 0.40000 0.60000 1.66667 0.09812 0.03532
6 0.05887 2 2 2 0.40000 0.60000 1.66667 0.09812 0.03532
6 0.05102 2 2 2 0.40000 0.60000 1.66667 0.08504 0.03061
6 0.05567 2 2 2 0.40000 0.60000 1.66667 0.09278 0.03340
6 0.04824 2 2 2 0.40000 0.60000 1.66667 0.08041 0.02895
6 0.04074 2 2 2 0.40000 0.60000 1.66667 0.06791 0.02445
6 0.05887 2 2 2 0.40000 0.60000 1.66667 0.09812 0.03532
6 0.05567 2 2 2 0.40000 0.60000 1.66667 0.09278 0.03340
6 0.00501 2 4 0 0.00000 1.00000 1.00000 0.00501 0.00501
6 0.00309 2 4 0 0.00000 1.00000 1.00000 0.00309 0.00309
6 0.00413 2 4 0 0.00000 1.00000 1.00000 0.00413 0.00413
6 0.01193 2 4 0 0.00000 1.00000 1.00000 0.01193 0.01193
6 0.00369 2 4 0 0.00000 1.00000 1.00000 0.00369 0.00369
6 0.00276 2 4 0 0.00000 1.00000 1.00000 0.00276 0.00276
6 0.01093 2 4 0 0.00000 1.00000 1.00000 0.01093 0.01093
6 0.00530 2 4 0 0.00000 1.00000 1.00000 0.00530 0.00530
6 0.01262 2 4 0 0.00000 1.00000 1.00000 0.01262 0.01262
6 0.01262 2 4 0 0.00000 1.00000 1.00000 0.01262 0.01262
6 0.00795 2 4 0 0.00000 1.00000 1.00000 0.00795 0.00795
6 0.00841 2 4 0 0.00000 1.00000 1.00000 0.00841 0.00841
6 0.00309 2 4 0 0.00000 1.00000 1.00000 0.00309 0.00309
6 0.00561 2 4 0 0.00000 1.00000 1.00000 0.00561 0.00561
6 0.00841 2 4 0 0.00000 1.00000 1.00000 0.00841 0.00841
6 0.00651 2 4 0 0.00000 1.00000 1.00000 0.00651 0.00651
6 0.00413 2 4 0 0.00000 1.00000 1.00000 0.00413 0.00413
6 0.00561 2 4 0 0.00000 1.00000 1.00000 0.00561 0.00561
6 0.00373 2 4 0 0.00000 1.00000 1.00000 0.00373 0.00373
6 0.00295 2 4 0 0.00000 1.00000 1.00000 0.00295 0.00295
6 0.00520 4 0 2 0.00000 1.00000 1.00000 0.00520 0.00520
6 0.00651 4 0 2 0.00000 1.00000 1.00000 0.00651 0.00651
6 0.00873 4 0 2 0.00000 1.00000 1.00000 0.00873 0.00873
6 0.01034 4 0 2 0.00000 1.00000 1.00000 0.01034 0.01034
6 0.01093 4 0 2 0.00000 1.00000 1.00000 0.01093 0.01093
6 0.01262 4 0 2 0.00000 1.00000 1.00000 0.01262 0.01262
6 0.00346 4 0 2 0.00000 1.00000 1.00000 0.00346 0.00346
6 0.01193 4 0 2 0.00000 1.00000 1.00000 0.01193 0.01193
6 0.00615 4 0 2 0.00000 1.00000 1.00000 0.00615 0.00615
6 0.01262 4 0 2 0.00000 1.00000 1.00000 0.01262 0.01262
6 0.01262 4 2 0 0.00000 1.00000 1.00000 0.01262 0.01262
6 0.00795 4 2 0 0.00000 1.00000 1.00000 0.00795 0.00795
6 0.00346 4 2 0 0.00000 1.00000 1.00000 0.00346 0.00346
6 0.00276 4 2 0 0.00000 1.00000 1.00000 0.00276 0.00276
6 0.00841 4 2 0 0.00000 1.00000 1.00000 0.00841 0.00841
6 0.00413 4 2 0 0.00000 1.00000 1.00000 0.00413 0.00413
6 0.00369 4 2 0 0.00000 1.00000 1.00000 0.00369 0.00369
6 0.01093 4 2 0 0.00000 1.00000 1.00000 0.01093 0.01093
6 0.00561 4 2 0 0.00000 1.00000 1.00000 0.00561 0.00561
6 0.00651 4 2 0 0.00000 1.00000 1.00000 0.00651 0.00651
6 0.00373 4 2 0 0.00000 1.00000 1.00000 0.00373 0.00373
6 0.00873 4 2 0 0.00000 1.00000 1.00000 0.00873 0.00873
6 0.00333 4 2 0 0.00000 1.00000 1.00000 0.00333 0.00333
6 0.00309 4 2 0 0.00000 1.00000 1.00000 0.00309 0.00309
6 0.00229 4 2 0 0.00000 1.00000 1.00000 0.00229 0.00229
6 0.01193 4 2 0 0.00000 1.00000 1.00000 0.01193 0.01193
6 0.00289 4 2 0 0.00000 1.00000 1.00000 0.00289 0.00289
6 0.00561 4 2 0 0.00000 1.00000 1.00000 0.00561 0.00561
6 0.00841 4 2 0 0.00000 1.00000 1.00000 0.00841 0.00841
6 0.00295 4 2 0 0.00000 1.00000 1.00000 0.00295 0.00295
6 0.01262 4 2 0 0.00000 1.00000 1.00000 0.01262 0.01262
6 0.00473 4 2 0 0.00000 1.00000 1.00000 0.00473 0.00473
6 0.00501 4 2 0 0.00000 1.00000 1.00000 0.00501 0.00501
6 0.00530 4 2 0 0.00000 1.00000 1.00000 0.00530 0.00530
6 0.01034 4 2 0 0.00000 1.00000 1.00000 0.01034 0.01034
6 0.00413 4 2 0 0.00000 1.00000 1.00000 0.00413 0.00413
6 0.00615 4 2 0 0.00000 1.00000 1.00000 0.00615 0.00615
6 0.00264 4 2 0 0.00000 1.00000 1.00000 0.00264 0.00264
6 0.00309 4 2 0 0.00000 1.00000 1.00000 0.00309 0.00309
6 0.00520 4 2 0 0.00000 1.00000 1.00000 0.00520 0.00520
6 0.00023 6 0 0 0.00000 1.00000 1.00000 0.00023 0.00023
6 0.00035 6 0 0 0.00000 1.00000 1.00000 0.00035 0.00035
6 0.00040 6 0 0 0.00000 1.00000 1.00000 0.00040 0.00040
6 0.00028 6 0 0 0.00000 1.00000 1.00000 0.00028 0.00028
6 0.00042 6 0 0 0.00000 1.00000 1.00000 0.00042 0.00042
6 0.00033 6 0 0 0.00000 1.00000 1.00000 0.00033 0.00033
6 0.00028 6 0 0 0.00000 1.00000 1.00000 0.00028 0.00028
6 0.00042 6 0 0 0.00000 1.00000 1.00000 0.00042 0.00042
6 0.00062 6 0 0 0.00000 1.00000 1.00000 0.00062 0.00062
6 0.00062 6 0 0 0.00000 1.00000 1.00000 0.00062 0.00062
6 0.00057 6 0 0 0.00000 1.00000 1.00000 0.00057 0.00057
6 0.00074 6 0 0 0.00000 1.00000 1.00000 0.00074 0.00074
6 0.00074 6 0 0 0.00000 1.00000 1.00000 0.00074 0.00074
6 0.00032 6 0 0 0.00000 1.00000 1.00000 0.00032 0.00032
6 0.00044 6 0 0 0.00000 1.00000 1.00000 0.00044 0.00044
6 0.00060 6 0 0 0.00000 1.00000 1.00000 0.00060 0.00060
6 0.00078 6 0 0 0.00000 1.00000 1.00000 0.00078 0.00078
6 0.00027 6 0 0 0.00000 1.00000 1.00000 0.00027 0.00027
6 0.00078 6 0 0 0.00000 1.00000 1.00000 0.00078 0.00078
6 0.00018 6 0 0 0.00000 1.00000 1.00000 0.00018 0.00018
Comparison with simulation:
The situation is simulated by:
DECLARE SUB returnSock (s#)
DEFDBL A-Z
RANDOMIZE TIMER
FOR week = 1 TO 5000
socks = 16
blue = 8: black = 6: brown = 2
FOR day = 1 TO 6
tryct = 0: match = 0
WHILE match = 0
r1 = INT(socks * RND(1) + 1)
DO
r2 = INT(socks * RND(1) + 1)
LOOP UNTIL r2 <> r1
DO
r3 = INT(socks * RND(1) + 1)
LOOP UNTIL r3 <> r1 AND r3 <> r2
IF r1 <= blue THEN
c1 = 1
ELSEIF r1 <= blue + black THEN
c1 = 2
ELSE
c1 = 3
END IF
IF r2 <= blue THEN
c2 = 1
ELSEIF r2 <= blue + black THEN
c2 = 2
ELSE
c2 = 3
END IF
IF r3 <= blue THEN
c3 = 1
ELSEIF r3 <= blue + black THEN
c3 = 2
ELSE
c3 = 3
END IF
SELECT CASE c1
CASE 1: blue = blue - 1
CASE 2: black = black - 1
CASE 3: brown = brown - 1
END SELECT
SELECT CASE c2
CASE 1: blue = blue - 1
CASE 2: black = black - 1
CASE 3: brown = brown - 1
END SELECT
SELECT CASE c3
CASE 1: blue = blue - 1
CASE 2: black = black - 1
CASE 3: brown = brown - 1
END SELECT
tryct = tryct + 1
IF c1 = c2 THEN
returnSock c3: match = 1
ELSEIF c1 = c3 THEN
returnSock c2: match = 1
ELSEIF c2 = c3 THEN
returnSock c1: match = 1
ELSE
returnSock c1: returnSock c2: returnSock c3
END IF
IF day = 1 THEN PRINT c1; c2; c3
WEND
socks = socks - 2
tries(day) = tries(day) + tryct
IF tryct = 1 THEN onetry(day) = onetry(day) + 1
IF day = 1 THEN
PRINT tryct, blue; black; brown, tries(day) / week
'
END IF
NEXT day
NEXT week
week = week - 1
FOR i = 1 TO 6
PRINT USING "###.#####"; tries(i) / week; onetry(i) / week
NEXT: PRINT
SUB returnSock (s)
SHARED blue, black, brown
SELECT CASE s
CASE 1: blue = blue + 1
CASE 2: black = black + 1
CASE 3: brown = brown + 1
END SELECT
END SUB
which simulates 5000 weeks--almost 100 years (95.8 years to be almost exact). The results (average number of tries to get a pair) of one particular run, as the random generator is randomized by the timer for a particular run, are:
avg length avg. prob.
1.20640 0.83000
1.22860 0.81520
1.24380 0.80460
1.30080 0.77940
1.32280 0.77140
1.35420 0.78880
in good agreement with the calculated probabilities.
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Posted by Charlie
on 2013-06-28 18:57:50 |
computer-assisted solution
