A player draws from a bag containing 10 black and 10 white marbles one by one, without putting them back in.
Every time before drawing a marble he guesses the color of the marble he will draw.
He decides to always guess the color that occurs most frequently remaining in the bag (if there is a tie, he chooses either color).
Find the expected number of correct guesses.
Find a formula for the number of correct guesses for starting with X black and Y white marbles.
This problem is a modified version of Guessing suit and addresses an interesting question that came up in the comments.
The expected number for 10 white and 10 black marbles is 1139735 / 92378 ~= 12.3377319275152.
Starting with UBASIC, as it can do rational numbers: We build up a table so that the probability for higher numbers of marbles can be based on the probabilities of smaller numbers and the probability that the larger number will go to the particular smaller numbers.
4 kill "bestgues.txt"
5 open "bestgues.txt" for output as #2
10 dim State(14,14)
20 for I=0 to 14:State(I,0)=I:next
30 for I=0 to 14:State(0,I)=I:next
40 for Tot=2 to 28
50 for White=1 to 14
60 Black=Tot-White
70 if Black<=14 and Black>=0 then
80 :if Black>White then
85 :if White>0 then
90 :State(White,Black)=(White//(White+Black))*State(White-1,Black)+(Black//(White+Black))*(State(White,Black-1)+1)
91 :else
92 :State(White,Black)=(Black//(White+Black))*(State(White,Black-1)+1)
95 :endif
100 :else
105 :if Black>0 then
110 :State(White,Black)=(Black//(White+Black))*State(White,Black-1)+(White//(White+Black))*(State(White-1,Black)+1)
111 :else
112 :State(White,Black)=(White//(White+Black))*(State(White-1,Black)+1)
115 :endif
120 :endif
130 next White
140 next Tot
150 for White=0 to 14
160 for Black=0 to 14
170 print State(White,Black);
172 print #2,State(White,Black);
180 next
190 print:print
192 print #2,:print #2,
200 next
205 close #2
Each row below represents a different number of white marbles from 0 to 14. In each row, the expected values are in the order from zero black marbles to 14 black marbles. If one of the colors has zero representation in the set of marbles, then of course, you get all the marbles, as shown on the first line and the first entry in each line. The doubled slashes of UBASIC have been replaced by the single slashes commonly associated with fractions:
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
1 3/2 7/3 13/4 21/5 31/6 43/7 57/8 73/9 91/10 111/11 133/12 157/13 183/14 211/15
2 7/3 17/6 18/5 67/15 113/21 177/28 131/18 371/45 507/55 673/66 436/39 1107/91 1381/105 1697/120
3 13/4 18/5 41/10 169/35 317/56 275/42 112/15 1387/165 2059/220 1476/143 2055/182 5581/455 7417/560 4837/340
4 21/5 67/15 169/35 373/70 380/63 718/105 1271/165 4259/495 6813/715 10480/1001 5197/455 22537/1820 15887/1190 10957/765
5 31/6 113/21 317/56 380/63 823/126 1667/231 3169/396 11389/1287 19489/2002 10657/1001 16855/1456 38735/3094 14429/1071 41957/2907
6 43/7 177/28 275/42 718/105 1667/231 3565/462 3598/429 27497/3003 49989/5005 86965/8008 72769/6188 58846/4641 92345/6783 28217/1938
7 57/8 131/18 112/15 1271/165 3169/396 3598/429 7625/858 61429/6435 117853/11440 108129/9724 95311/7956 162113/12597 53411/3876 42752/2907
8 73/9 371/45 1387/165 4259/495 11389/1287 27497/3003 61429/6435 129293/12870 130008/12155 250292/21879 462793/37791 164962/12597 284381/20349 475738/31977
9 91/10 507/55 2059/220 6813/715 19489/2002 49989/5005 117853/11440 130008/12155 272171/24310 546773/46189 211151/16796 392909/29393 706683/49742 1232163/81719
10 111/11 673/66 1476/143 10480/1001 10657/1001 86965/8008 108129/9724 250292/21879 546773/46189 1139735/92378 1143934/88179 4428771/323323 8285509/572033 5006119/326876
11 133/12 436/39 2055/182 5197/455 16855/1456 72769/6188 95311/7956 462793/37791 211151/16796 1143934/88179 2376047/176358 9533581/676039 6165043/416024 5793593/371450
12 157/13 1107/91 5581/455 22537/1820 38735/3094 58846/4641 162113/12597 164962/12597 392909/29393 4428771/323323 9533581/676039 19743201/1352078 19795204/1300075 38476058/2414425
13 183/14 1381/105 7417/560 15887/1190 14429/1071 92345/6783 53411/3876 284381/20349 706683/49742 8285509/572033 6165043/416024 19795204/1300075 40890483/2600150 81966691/5014575
14 211/15 1697/120 4837/340 10957/765 41957/2907 28217/1938 42752/2907 475738/31977 1232163/81719 5006119/326876 5793593/371450 38476058/2414425 81966691/5014575 168947957/10029150
For decimal probabilities, in a more neat format, switching to Visual Basic 5.0. The VB version also includes a simulation for 10,10 and 3.6:
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
0 0.00000 1.00000 2.00000 3.00000 4.00000 5.00000 6.00000 7.00000 8.00000 9.00000 10.00000 11.00000 12.00000 13.00000 14.00000
1 1.00000 1.50000 2.33333 3.25000 4.20000 5.16667 6.14286 7.12500 8.11111 9.10000 10.09091 11.08333 12.07692 13.07143 14.06667
2 2.00000 2.33333 2.83333 3.60000 4.46667 5.38095 6.32143 7.27778 8.24444 9.21818 10.19697 11.17949 12.16484 13.15238 14.14167
3 3.00000 3.25000 3.60000 4.10000 4.82857 5.66071 6.54762 7.46667 8.40606 9.35909 10.32168 11.29121 12.26593 13.24464 14.22647
4 4.00000 4.20000 4.46667 4.82857 5.32857 6.03175 6.83810 7.70303 8.60404 9.52867 10.46953 11.42198 12.38297 13.35042 14.32288
5 5.00000 5.16667 5.38095 5.66071 6.03175 6.53175 7.21645 8.00253 8.84926 9.73477 10.64635 11.57624 12.51939 13.47246 14.43309
6 6.00000 6.14286 6.32143 6.54762 6.83810 7.21645 7.71645 8.38695 9.15651 9.98781 10.85977 11.75970 12.67959 13.61418 14.55986
7 7.00000 7.12500 7.27778 7.46667 7.70303 8.00253 8.38695 8.88695 9.54608 10.30184 11.11981 11.97976 12.86918 13.77993 14.70657
8 8.00000 8.11111 8.24444 8.40606 8.60404 8.84926 9.15651 9.54608 10.04608 10.69585 11.43983 12.24612 13.09534 13.97518 14.87751
9 9.00000 9.10000 9.21818 9.35909 9.52867 9.73477 9.98781 10.30184 10.69585 11.19585 11.83773 12.57151 13.36743 14.20697 15.07805
10 10.00000 10.09091 10.19697 10.32168 10.46953 10.64635 10.85977 11.11981 11.43983 11.83773 12.33773 12.97286 13.69767 14.48432 15.31504
11 11.00000 11.08333 11.17949 11.29121 11.42198 11.57624 11.75970 11.97976 12.24612 12.57151 12.97286 13.47286 14.10212 14.81896 15.59724
12 12.00000 12.07692 12.16484 12.26593 12.38297 12.51939 12.67959 12.86918 13.09534 13.36743 13.69767 14.10212 14.60212 15.22620 15.93591
13 13.00000 13.07143 13.15238 13.24464 13.35042 13.47246 13.61418 13.77993 13.97518 14.20697 14.48432 14.81896 15.22620 15.72620 16.34569
14 14.00000 14.06667 14.14167 14.22647 14.32288 14.43309 14.55986 14.70657 14.87751 15.07805 15.31504 15.59724 15.93591 16.34569 16.84569
12.2451 (Simulation for 10,10
6.53525 and then for 3,6; 10,000 trials each)
DefDbl A-Z
Dim crlf$
Dim State(14, 14)
Function mform$(x, t$)
a$ = Format$(x, t$)
If Len(a$) < Len(t$) Then a$ = Space$(Len(t$) - Len(a$)) & a$
mform$ = a$
End Function
Private Sub Form_Load()
Form1.Visible = True
Text1.Text = ""
crlf = Chr$(13) + Chr$(10)
For I = 0 To 14: State(I, 0) = I: Next
For I = 0 To 14: State(0, I) = I: Next
For Tot = 2 To 28
For white = 1 To 14
black = Tot - white
If black <= 14 And black >= 0 Then
If black > white Then
If white > 0 Then
State(white, black) = (white / (white + black)) * State(white - 1, black) + (black / (white + black)) * (State(white, black - 1) + 1)
Else
State(white, black) = (black / (white + black)) * (State(white, black - 1) + 1)
End If
Else
If black > 0 Then
State(white, black) = (black / (white + black)) * State(white, black - 1) + (white / (white + black)) * (State(white - 1, black) + 1)
Else
State(white, black) = (white / (white + black)) * (State(white - 1, black) + 1)
End If
End If
End If
Next white
Next Tot
For white = 0 To 14
For black = 0 To 14
Text1.Text = Text1.Text & mform(State(white, black), "##0.00000")
Next
Text1.Text = Text1.Text & crlf
Next
For tr = 1 To 10000
white = 10: black = 10
Randomize Timer
For draw = 1 To 20
r = Int(Rnd(1) * (white + black) + 1)
If r <= white Then
If white > black Then
hits = hits + 1:
ElseIf white = black Then
hits = hits + 0.5
End If
white = white - 1
Else
If black > white Then
hits = hits + 1:
ElseIf white = black Then
hits = hits + 0.5
End If
black = black - 1
End If
Next draw
Next tr
tr = tr - 1
Text1.Text = Text1.Text & hits / tr & crlf
hits = 0
For tr = 1 To 10000
white = 3: black = 6
Randomize Timer
For draw = 1 To 9
r = Int(Rnd(1) * (white + black) + 1)
If r <= white Then
If white > black Then
hits = hits + 1:
ElseIf white = black Then
hits = hits + 0.5
End If
white = white - 1
Else
If black > white Then
hits = hits + 1:
ElseIf white = black Then
hits = hits + 0.5
End If
black = black - 1
End If
Next draw
Next tr
tr = tr - 1
Text1.Text = Text1.Text & hits / tr & crlf
Text1.Text = Text1.Text & crlf & " done"
End Sub
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Posted by Charlie
on 2015-07-30 14:56:26 |
computer solution
