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Rolling to score (Posted on 2010-11-15) Difficulty: 4 of 5
We roll five standard dice (sides numbered 1 to 6) and write down the sum of the top three i.e. of the 3 highest values.
What is the probability to get 15 ?

No Solution Yet Submitted by Ady TZIDON    
Rating: 3.3333 (3 votes)

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Solution computer-aided solution | Comment 4 of 17 |

The following program examines each possible combination of numbers of the various numbers of pips, using each one's probability to add to the corresponding total's probability of occurring:

  100       Ndice=5:Ntop=3
  110       Remaining=Ndice:CurrProb=1
  120       dim TProb(Ndice*6)
  130       dim Hist(6)
  140 
  150       gosub *SetQuant(6)
  160       for I=1 to Ndice*6
  170          CProb=CProb+TProb(I)
  180          print I,TProb(I),CProb/1,1-CProb/1
  190       next I
  200 
  210       end
  220 
 1100           *SetQuant(Den)
 1110           local Nbr,P,Strt
 1120           if Den=1 then Strt=Remaining:else Strt=0
 1130           for Nbr=Strt to Remaining
 1135             if Den=1 then
 1136               :P=1
 1137             :else
 1140               :P=(1//Den)^Nbr*((Den-1)//Den)^(Remaining-Nbr)*combi(Remaining,Nbr)
 1150          
 1160             CurrProb=CurrProb*P
 1170             Remaining=Remaining-Nbr
 1180             Hist(Den)=Nbr
 1190          
 1200             if Remaining=0 then
 1210              :gosub *CountIt(Den)
 1220             :else
 1230              :gosub *SetQuant(Den-1)
 1240          
 1250             Remaining=Remaining+Nbr
 1260             CurrProb=CurrProb//P
 1270           next Nbr
 1280           return
 1290        
 2100           *CountIt(D)
 2110           local Pips,Totv,Totn
 2120           for Pips=6 to D step -1
 2130             if Hist(Pips)>=Ntop-Totn then
 2140              :Totv=Totv+Pips*(Ntop-Totn)
 2150              :Totn=Ntop
 2160             :else
 2170              :Totv=Totv+Pips*Hist(Pips)
 2180              :Totn=Totn+Hist(Pips)
 2190           next Pips
 2200           TProb(Totv)=TProb(Totv)+CurrProb
 2210           return
 
 

The result is a display that shows all the possible totals:


 
score   p(score)        p(<= this score)        p(> this score)
 1      0               0                       1
 2      0               0                       1
 3      1/7776         0.0001286008230452674   0.9998713991769547325
 4      5/7776         0.0007716049382716048   0.9992283950617283951
 5      5/2592         0.0027006172839506172   0.9972993827160493827
 6      41/7776        0.0079732510288065843   0.9920267489711934156
 7      5/432          0.0195473251028806584   0.9804526748971193415
 8      85/3888        0.0414094650205761316   0.9585905349794238683
 9      37/972         0.0794753086419753086   0.9205246913580246913
10      235/3888       0.1399176954732510287   0.8600823045267489712
11      665/7776       0.2254372427983539094   0.7745627572016460905
12      881/7776       0.3387345679012345678   0.6612654320987654321
13      1055/7776      0.4744084362139917695   0.5255915637860082304
14      385/2592       0.6229423868312757201   0.3770576131687242798
15      1111/7776      0.7658179012345679012   0.2341820987654320987
16      935/7776       0.886059670781893004    0.1139403292181069959
17      305/3888       0.9645061728395061728   0.0354938271604938271
18      23/648         1.0                     0
19      0               1.0                     0
20      0               1.0                     0
21      0               1.0                     0
22      0               1.0                     0
23      0               1.0                     0
24      0               1.0                     0
25      0               1.0                     0
26      0               1.0                     0
27      0               1.0                     0
28      0               1.0                     0
29      0               1.0                     0
30      0               1.0                     0

So the probability of a score of exactly 15 is 1111/7776 ~= .1428755144032922. The probability of getting at least 15 is the same as the probability of getting > 14, which is shown above as being approximately 0.3770576131687242798. Adding the individual probabilities for 15 through 18 gives this as exactly 733/1944.

As all the denominators can be expanded to 6^5 = 7776, the following table can be made by a slight modification of the program, in which the last two columns have been multiplied by 7776, so as to provide the numerators for a fraction with that number in the denominator:

value    p(score=value) p(score<=value) p(score > value)
                         -------numerators only -------
 2       0              0                 7776
 3       1/7776         1                 7775
 4       5/7776         6                 7770
 5       5/2592         21                7755
 6       41/7776        62                7714
 7       5/432          152               7624
 8       85/3888        322               7454
 9       37/972         618               7158
 10      235/3888       1088              6688
 11      665/7776       1753              6023
 12      881/7776       2634              5142
 13      1055/7776      3689              4087
 14      385/2592       4844              2932
 15      1111/7776      5955              1821
 16      935/7776       6890              886
 17      305/3888       7500              276
 18      23/648         7776              0
 
 
 

As the specific interest in this problem is for a score of 15, the following details the derivation of probability 1111/7776 for that score. The sets of showing numbers of pips are all those that score 15. The first probability, for example, is 5/3888 = 10/7776, as 3 5's and two 1's can occur in C(5,3) = C(5,2) = 10 ways.
 

     number of 
 6's  5's  4's  3's  2's  1's     probability
  0    3    0    0    0    2        5/3888
  0    3    0    0    1    1        5/1944
  0    3    0    0    2             5/3888
  0    3    0    1    0    1        5/1944
  0    3    0    1    1             5/1944
  0    3    0    2                  5/3888
  0    3    1    0    0    1        5/1944
  0    3    1    0    1             5/1944
  0    3    1    1                  5/1944
  0    3    2                       5/3888
  0    4    0    0    0    1        5/7776
  0    4    0    0    1             5/7776
  0    4    0    1                  5/7776
  0    4    1                       5/7776
  0    5                            1/7776
  1    1    1    0    0    2        5/648
  1    1    1    0    1    1        5/324
  1    1    1    0    2             5/648
  1    1    1    1    0    1        5/324
  1    1    1    1    1             5/324
  1    1    1    2                  5/648
  1    1    2    0    0    1        5/648
  1    1    2    0    1             5/648
  1    1    2    1                  5/648
  1    1    3                       5/1944
  2    0    0    1    0    2        5/1296
  2    0    0    1    1    1        5/648
  2    0    0    1    2             5/1296
  2    0    0    2    0    1        5/1296
  2    0    0    2    1             5/1296
  2    0    0    3                  5/3888
 
which do indeed add up to 1111/7776. 

Simulation verification:

DEFDBL A-Z
CLS
DIM valCt(20)
FOR trial = 1 TO 1000000
  tot = 0
  FOR die = 1 TO 5
    pips(die) = INT(RND(1) * 6) + 1
  NEXT die
  DO
    done = 1
    FOR i = 1 TO 4
      IF pips(i) < pips(i + 1) THEN SWAP pips(i), pips(i + 1): done = 0
    NEXT
  LOOP UNTIL done = 1
  FOR i = 1 TO 3
    tot = tot + pips(i)
  NEXT
  valCt(tot) = valCt(tot) + 1
NEXT trial

FOR i = 3 TO 18
  c = c + valCt(i) / (trial - 1)
  PRINT USING "## #.####### #.#######"; i; valCt(i) / (trial - 1); c
NEXT

goes through a million trials and gets the following statistics:

val  fraction =val   fraction <= val
 3   0.000142           0.000142
 4   0.000625           0.000767
 5   0.001896           0.002663
 6   0.005248           0.007911
 7   0.011541           0.019452
 8   0.021726           0.041178
 9   0.038271           0.079449
10   0.060519           0.139968
11   0.085300           0.225268
12   0.113187           0.338455
13   0.135842           0.474297
14   0.148759           0.623056
15   0.142818           0.765874
16   0.120569           0.886443
17   0.077907           0.964350
18   0.035650           1.000000


 


  Posted by Charlie on 2010-11-15 18:33:10
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