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 Even (related) chances (Posted on 2017-08-09)
Consider a string of n quaternary digits, randomly created.

Evaluate the probability that in this string the quantity
of an a priori defined digit, say zero, is even.

either
a. direct count and comparison (e.g. n=4)
or
b. getting the same result by two distinct methods.

 No Solution Yet Submitted by Ady TZIDON No Rating

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 solution (spoiler) | Comment 1 of 4
Sigma{i=0 to floor(n/2)} (1/4)^(2i) * (3/4)^(n-2i) * C(n,2i)

` n  probability 1 .75 2 .625 3 .5625 4 .53125 5 .515625 6 .5078125 7 .50390625 8 .501953125 9 .500976562510 .5004882812511 .50024414062512 .500122070312513 .5000610351562514 .50003051757812515 .50001525878906316 .50000762939453117 .50000381469726618 .50000190734863319 .50000095367431620 .50000047683715821 .50000023841857922 .5000001192092923 .50000005960464524 .50000002980232225 .500000014901161`

from

Form1.Visible = True

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

For n = 1 To 25
p = 0
For i = 0 To n / 2
p = p + (1 / 4) ^ (2 * i) * (3 / 4) ^ (n - 2 * i) * combi(n, 2 * i)
Next i
Text1.Text = Text1.Text & n & Str(p) & crlf
Next

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

End Sub

Function combi(n, r)
c = 1
For i = n To n - r + 1 Step -1
c = c * i
Next
For i = 2 To r
c = c / i
Next
combi = c
End Function

Double check via choice a:

direct count of even number of zeros (evens marked with *):

0000   4 *
0001   3
0002   3
0003   3
0010   3
0011   2 *
0012   2 *
0013   2 *
0020   3
0021   2 *
0022   2 *
0023   2 *
0030   3
0031   2 *
0032   2 *
0033   2 *
0100   3
0101   2 *
0102   2 *
0103   2 *
0110   2 *
0111   1
0112   1
0113   1
0120   2 *
0121   1
0122   1
0123   1
0130   2 *
0131   1
0132   1
0133   1
0200   3
0201   2 *
0202   2 *
0203   2 *
0210   2 *
0211   1
0212   1
0213   1
0220   2 *
0221   1
0222   1
0223   1
0230   2 *
0231   1
0232   1
0233   1
0300   3
0301   2 *
0302   2 *
0303   2 *
0310   2 *
0311   1
0312   1
0313   1
0320   2 *
0321   1
0322   1
0323   1
0330   2 *
0331   1
0332   1
0333   1
1000   3
1001   2 *
1002   2 *
1003   2 *
1010   2 *
1011   1
1012   1
1013   1
1020   2 *
1021   1
1022   1
1023   1
1030   2 *
1031   1
1032   1
1033   1
1100   2 *
1101   1
1102   1
1103   1
1110   1
1111   0 *
1112   0 *
1113   0 *
1120   1
1121   0 *
1122   0 *
1123   0 *
1130   1
1131   0 *
1132   0 *
1133   0 *
1200   2 *
1201   1
1202   1
1203   1
1210   1
1211   0 *
1212   0 *
1213   0 *
1220   1
1221   0 *
1222   0 *
1223   0 *
1230   1
1231   0 *
1232   0 *
1233   0 *
1300   2 *
1301   1
1302   1
1303   1
1310   1
1311   0 *
1312   0 *
1313   0 *
1320   1
1321   0 *
1322   0 *
1323   0 *
1330   1
1331   0 *
1332   0 *
1333   0 *
2000   3
2001   2 *
2002   2 *
2003   2 *
2010   2 *
2011   1
2012   1
2013   1
2020   2 *
2021   1
2022   1
2023   1
2030   2 *
2031   1
2032   1
2033   1
2100   2 *
2101   1
2102   1
2103   1
2110   1
2111   0 *
2112   0 *
2113   0 *
2120   1
2121   0 *
2122   0 *
2123   0 *
2130   1
2131   0 *
2132   0 *
2133   0 *
2200   2 *
2201   1
2202   1
2203   1
2210   1
2211   0 *
2212   0 *
2213   0 *
2220   1
2221   0 *
2222   0 *
2223   0 *
2230   1
2231   0 *
2232   0 *
2233   0 *
2300   2 *
2301   1
2302   1
2303   1
2310   1
2311   0 *
2312   0 *
2313   0 *
2320   1
2321   0 *
2322   0 *
2323   0 *
2330   1
2331   0 *
2332   0 *
2333   0 *
3000   3
3001   2 *
3002   2 *
3003   2 *
3010   2 *
3011   1
3012   1
3013   1
3020   2 *
3021   1
3022   1
3023   1
3030   2 *
3031   1
3032   1
3033   1
3100   2 *
3101   1
3102   1
3103   1
3110   1
3111   0 *
3112   0 *
3113   0 *
3120   1
3121   0 *
3122   0 *
3123   0 *
3130   1
3131   0 *
3132   0 *
3133   0 *
3200   2 *
3201   1
3202   1
3203   1
3210   1
3211   0 *
3212   0 *
3213   0 *
3220   1
3221   0 *
3222   0 *
3223   0 *
3230   1
3231   0 *
3232   0 *
3233   0 *
3300   2 *
3301   1
3302   1
3303   1
3310   1
3311   0 *
3312   0 *
3313   0 *
3320   1
3321   0 *
3322   0 *
3323   0 *
3330   1
3331   0 *
3332   0 *
3333   0 *

asterisked lines show an even number of zeros

136 out of 256 = .53125

from

For a = 0 To 3
For b = 0 To 3
For c = 0 To 3
For d = 0 To 3
Text1.Text = Text1.Text & a & b & c & d & "   "
subct = 0
If a = 0 Then subct = subct + 1
If b = 0 Then subct = subct + 1
If c = 0 Then subct = subct + 1
If d = 0 Then subct = subct + 1
Text1.Text = Text1.Text & subct
If subct Mod 2 = 0 Then Text1.Text = Text1.Text & " *": ct = ct + 1
Text1.Text = Text1.Text & crlf
Next
Next
Next
Next

 Posted by Charlie on 2017-08-09 11:34:37

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