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Random^Random (Posted on 2014-10-29) Difficulty: 5 of 5
Let A and B each be random real numbers chosen from the uniform interval (0,1).

Call Z the tenths place digit of AB.

Find the probability distribution of Z.

See The Solution Submitted by Jer    
Rating: 5.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Some Thoughts re: towards an analytic solution - numeric integration | Comment 3 of 9 |
(In reply to towards an analytic solution by Charlie)

Actually that would be 1 -  Integral{0 to 1} ln(.9)/ln(A) dA.


It comes out to 0.2870992757... (this is for digit 9) per the .9 in the formula.

via:

DefDbl A-Z
Dim crlf$
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()
 ChDir "C:\Program Files (x86)\DevStudio\VB\projects\flooble"
 Text1.Text = ""
 crlf$ = Chr(13) + Chr(10)
 Form1.Visible = True
 DoEvents
 
 lp9 = Log(0.9)
 n = 100000
 For x = 1 / (2 * n) To 1 - 1 / (2 * n) Step 1 / n
  tot = tot + Exp(lp9 / x)
 Next
 DoEvents

 Text1.Text = Text1.Text & crlf
 Text1.Text = Text1.Text & crlf
 n = 100000
  tot = 0
 For x = 1 / (2 * n) To 1 - 1 / (2 * n) Step 1 / n
  tot = tot + Exp(lp9 / x)
 Next
 Text1.Text = Text1.Text & x & " " & 1 - tot / n

 
 Text1.Text = Text1.Text & crlf
 Text1.Text = Text1.Text & crlf
 n = 10000000
  tot = 0
 For x = 1 / (2 * n) To 1 - 1 / (2 * n) Step 1 / n
  tot = tot + Exp(lp9 / x)
 Next
 Text1.Text = Text1.Text & x & " " & 1 - tot / n
 
 
 Text1.Text = Text1.Text & crlf & "done"
End Sub

showing


 0.28709927571142

 0.287099275716361

as approximations using increments of 1/100,000 and 1/10,000,000 respectively.

  Posted by Charlie on 2014-10-29 15:32:44
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