A bag has 5 tokens numbered 1 to 5. Tokens are drawn with replacement until a token is repeated. What is the probability that the repeated token is the first token drawn?
What is the formula for N tokens?
At the second draw, the probability of repeating the first draw is 1/5.
The probability that the second draw does not match but the third does is (4/5)*(1/5) = 4/25.
The probability that the second draw doesn't match the first and the third matches neither, but the fourth matches the first is (4/5)*(3/5)*(1/5).
The probability that none of the first four match but the 5th matches the first is (4/5)*(3/5)*(2/5)*(1/5)
The probability that none of the first five match but the 6th matches the first is (4/5)*(3/5)*(2/5)*(1/5)*(1/5).
The total of these is what's sought for part 1. That's 1569/3125.
In general:
Sigma{i=1 to n} n!/((n-i)!*n^(i+1))
A series of 100,000 simulation trials resulted in 50,106 successes:
50106 / 100000 = .50106
* 3125 = 1565.8125
also showing the proportion relative to 3125, the denominator of the calculated probability, is 1566 (rounded), which is close to the numerator.
A table of probabilities, based on the formula is:
n probability
2 .75
3 .62962962962963
4 .5546875
5 .50208
6 .462448559670782
7 .431162671530206
8 .405627250671387
9 .384257304987295
10 .366021568
11 .350215640976124
12 .336339472924913
13 .324026762534809
14 .31300210174168
15 .303053819009815
16 .294016140442042
17 .285757107105978
18 .278170172166272
19 .271168221103024
20 .264679229300045
21 .258643051588228
22 .253009010357279
23 .247734057345501
24 .242781354330108
25 .23811916424981
26 .233719975487927
27 .22955980343996
28 .225617628405121
29 .221874939396167
30 .218315361034434
31 .214924346199219
32 .211688921145929
33 .20859747281408
34 .205639570303766
35 .202805814210464
36 .200087708816626
37 .197477553147685
38 .194968347684308
39 .192553714136775
40 .190227826171372
41 .187985349362805
42 .185821388953309
43 .183731444245465
44 .181711368654649
45 .179757334608565
46 .177865802613113
47 .176033493911919
48 .174257366255849
49 .172534592372509
50 .170862540786879
finally a randomized set of 100,000 trials for n = 50:
16987 / 100000 = .16987
DefDbl A-Z
Dim crlf$, had()
Private Sub Form_Load()
Form1.Visible = True
Text1.Text = ""
crlf = Chr$(13) + Chr$(10)
Randomize Timer
For tr = 1 To 100000
ReDim had(5)
r = Int(Rnd(1) * 5 + 1)
had(r) = 1
Do
r = Int(Rnd(1) * 5 + 1)
If had(r) Then
If had(r) = 1 Then succ = succ + 1
Exit Do
End If
had(r) = 2
Loop
Next
Text1.Text = Text1.Text & succ & " / 100000 = " & Str(succ / 100000) & crlf
Text1.Text = Text1.Text & " * 3125 = " & Str(succ * 3125 / 100000) & crlf
Text1.Text = Text1.Text & crlf
For n = 2 To 50
Text1.Text = Text1.Text & n
p = 0
For i = 1 To n
p = p + fact(n) / (fact(n - i) * n ^ (i + 1))
Next
Text1.Text = Text1.Text & Str(p) & crlf
Next
succ = 0
Randomize Timer
For tr = 1 To 100000
ReDim had(50)
r = Int(Rnd(1) * 50 + 1)
had(r) = 1
Do
r = Int(Rnd(1) * 50 + 1)
If had(r) Then
If had(r) = 1 Then succ = succ + 1
Exit Do
End If
had(r) = 2
Loop
Next
Text1.Text = Text1.Text & crlf
Text1.Text = Text1.Text & succ & " / 100000 = " & Str(succ / 100000) & crlf
Text1.Text = Text1.Text & crlf & " done"
End Sub
Function fact(x)
f = 1
For i = 1 To x
f = f * i
Next
fact = f
End Function
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Posted by Charlie
on 2016-07-18 15:04:40 |
solution
