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Starting the Scrabble Game (Posted on 2018-01-24) Difficulty: 2 of 5
To determine who plays first in a game of Scrabble, each player draws a tile and the one closest to A wins, except if one draws a blank tile. A blank tile beats any letter.

If there are only two players, what is the probability they both draw the same letter or both draw a blank?

The letters (and counts) are E (12); A, I (9); O (8); N, R, T (6); D, L, S, U (4); G (3); B, C, F, H, M, P, V, W, Y (2); J, K, Q, X, Z (1); and 2 blanks. -- 100 tiles in all.

Bonus: What if there are more than two players: what is the probability of a tie for who plays first? (subsequent play is to the left so 2nd, etc. player are not determined by the draw).

  Submitted by Charlie    
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Solution: (Hide)
There are 9900 ways the two players could pick their tiles.

The number of ways a particular number can contribute to the overall probability of getting two of the same letter is the count of how many there are of that letter times one less. The count table in the program below also has the number of letters that have that particular count, for example only one letter has 12 tiles with that letter on it; that accounts for the first line having entries of 12 (the count of that individual letter) and 1 (how many letters of the alphabet have that count of tiles).

Putting it into the succinct form from Jer's comment, it's

(12*11+9*8*2+8*7+6*5*3+4*3*4+3*2+2*1*10+1*0*5)/9900

with the 10 in 2*1*10 being the ten tiles that are duplicated, including the blank tiles.

The result comes out to

496/9900 = 124/2475 ~= 0.05010101010101

 ct(1, 0) = 12: ct(1, 1) = 1
 ct(2, 0) = 9: ct(2, 1) = 2
 ct(3, 0) = 8: ct(3, 1) = 1
 ct(4, 0) = 6: ct(4, 1) = 3
 ct(5, 0) = 4: ct(5, 1) = 4
 ct(6, 0) = 3: ct(6, 1) = 1
 ct(7, 0) = 2: ct(7, 1) = 10
 ct(8, 0) = 1: ct(8, 1) = 5

 For i = 1 To 8
   tot = tot + ct(i, 0) * ct(i, 1)
 Next
 Text1.Text = Text1.Text & tot & crlf & crlf

 For i = 1 To 8
   ways = ways + ct(i, 1) * ct(i, 0) * (ct(i, 0) - 1)
 Next
 Text1.Text = Text1.Text & ways & "/" & "9900 = " & Format(ways / 9900, "0.00000000")
 

For the bonus question the calculation becomes harder, as we need calculate not only whether some pair of players has a match, but whether that match is of the letter is the one that is closest to A (or blank). For example JCJ does not count as a tie as C is uniquely the winner, while CJC is a tie as the best scoring letter occurs twice.

As an example of the calculations, consider the case of 4 players, and in particular the contribution to the probability of a matching pair from the letter C.

There are 13 tiles lower (better) than C, including the two blank tiles. The probability that there are none of these tiles present is (87/100)*(86/99)*(85/98)*(84/97). There are 2 C tiles, so there are 15 tiles lower than D and the probability that none of these are chosen is (85/100)*(84/99)*(83/98)*(82/99). Subtracting the latter product from the former gives the probability that at least one C is present but nothing lower.

The probability of having exactly one C given nothing lower than C is present is 4*(2/87)*(85/86)*(84/85)*(83/84). The factor of 4 is the fact any of the 4 players could have chosen the sole C. This is subtracted from the probability having any C's given there's none under C, to get the conditional probability there's more than one C. Then that's multiplied by the probability that in fact there is nothing lower than C.

The program below calculates the probability first that for any given letter or blank, that no other letter has a letter closer to A or blank than itself, and then the probability, given that condition, that it will occur, and that it will not be unique. The overall probability of that happening is then added to the total for all the letters, and blank, that could be duplicated.

   5   open "scrblstr.txt" for output as #2
  10   dim tiles(26),before(27)
  20   tiles(0)=2
  30   tiles(1)=9
  40   tiles(2)=2
  50   tiles(3)=2
  60   tiles(4)=4
  70   tiles(5)=12
  80   tiles(6)=2
  90   tiles(7)=3
 100   tiles(8)=2
 110   tiles(9)=9
 120   tiles(10)=1
 130   tiles(11)=1
 140   tiles(12)=4
 150   tiles(13)=2
 160   tiles(14)=6
 170   tiles(15)=8
 180   tiles(16)=2
 190   tiles(17)=1
 200   tiles(18)=6
 210   tiles(19)=4
 220   tiles(20)=6
 230   tiles(21)=4
 240   tiles(22)=2
 250   tiles(23)=2
 260   tiles(24)=1
 270   tiles(25)=2
 280   tiles(26)=1
 290
 300   for i=1 to 26
 310     before(i)=before(i-1)+tiles(i-1)
 320   next
 321   before(27)=100
 330   print before(26)+tiles(26)
 340
 350   for noPlayers=2 to 8
 351     totProb=0
 360     for ltr=0 to 26
 370       probOnly=1
 380       numr=100-before(ltr):deno=100
 390       p=1
 400       repeat
 410         probOnly=probOnly*numr//deno
 420         dec numr:dec deno
 430         inc p
 440       until p>noPlayers or numr<0
 450       probNotHave=1  ' given that only this and above occur
 460       numr=100-before(ltr+1):deno=100-before(ltr)
 470       p=1
 480       repeat
 490         probNotHave=probNotHave*numr//deno
 500         dec numr:dec deno
 510         inc p
 520       until p>noPlayers or deno=0 or numr<0
 530       probHave=1-probNotHave  ' given that only this and above occur
 531       'if ltr<16 then print ltr,probHave/1
 540       if probHave>0 and ltr<26 then
 543         :probHave1= tiles(ltr)//(100-before(ltr))
 545         :p=2
 547         :numr=100-before(ltr+1):deno=99-before(ltr)
 549         :repeat
 551            :probHave1=probHave1*numr//deno
 553            :dec numr:dec deno
 555            :inc p
 557         :until p>noPlayers or deno=0 or numr<0
 559         :probHave1=probHave1*noPlayers
 560         :probHave1plus=probHave-probHave1
 570         :totProb=totProb+probOnly*probHave1plus
 580       :endif
 590     next ltr
 600     print noPlayers,totProb,totProb/1
 605     print #2,noPlayers,totProb,totProb/1
 610   next noPlayers
 620   close #2

The results are:

 2   124/2475                0.0501010101010101009
 3   25/294                  0.0850340136054421768
 4   456503/3921225          0.1164184661680979795
 5   3654367/25095840        0.1456164447972253568
 6   205978427/1192052400    0.1727930978537520665
 7   1585995557/8003780400   0.1981558060988279988
 8   187764172/845854065     0.2219817575742217423

A simulation of ten million trials shows agreement with this:

players    probability
2   501443/10000000 = 0.0501443
3   849717/10000000 = 0.0849717
4   1166902/10000000 = 0.1166902
5   1455121/10000000 = 0.1455121
6   1726306/10000000 = 0.1726306
7   1980379/10000000 = 0.1980379
8   2216635/10000000 = 0.2216635

DefDbl A-Z
Dim crlf$, age(4), digits$, ties(10), used()


Private Sub Form_Load()
 Form1.Visible = True


 Text1.Text = ""
 crlf = Chr$(13) + Chr$(10)

 s0$ = "EEEEEEEEEEEEAAAAAAAAAIIIIIIIIIOOOOOOOONNNNNNRRRRRRTTTTTTDDDDLLLLSSSSUUUU"
 s0$ = s0$ + "GGGBBCCFFHHMMPPVVWWYYJKQXZ  "
 Text1.Text = Text1.Text & Len(s0) & crlf & crlf

 For rpt = 1 To 100
 Randomize Timer
 For tr = 1 To 100000
    s$ = s0$
    lowest = 999
    ReDim used(32 To 96)
    For player = 1 To 8
      DoEvents
      r = 1 + Int(Rnd(1) * Len(s))
      v = Asc(Mid(s, r, 1))
      s = Left(s, r - 1) + Mid(s, r + 1)
      If v < lowest Then lowest = v
      used(v) = used(v) + 1
      If player > 1 Then
         If used(lowest) > 1 Then ties(player) = ties(player) + 1
      End If
    Next
    trials = trials + 1
 Next tr
 Next rpt

 For i = 2 To 8
   Text1.Text = Text1.Text & i & "   " & ties(i) & "/" & trials & " = " & ties(i) / trials & crlf
 Next

 Text1.Text = Text1.Text & crlf & " done"

End Sub

Comments: ( You must be logged in to post comments.)
  Subject Author Date
A factor of 10Larry2018-01-26 08:38:56
Solution3 people solutionJer2018-01-25 22:16:50
re(5): No SubjectJer2018-01-25 15:09:40
re(4): No Subjectarmando2018-01-25 05:22:48
re(3): No SubjectCharlie2018-01-24 21:47:57
re(2): No Subjectarmando2018-01-24 16:10:05
re: No SubjectCharlie2018-01-24 15:15:51
SolutionNo SubjectJer2018-01-24 14:47:34
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