Probability tricky question

[jack00123]

I am having trouble to figure out this tricky question. I kind of thought using the prob function but I guess I am wrong Pls help..

Four relatives are going to give you money for your birthday. The amount each relative gives you is equally likely to be any number between $0 and $200. The chance that the total amount of money you receive is at most $300 is ......?

 

[Gary's Student]

Around 20%

 

[jack00123]

Are you sure it is around 20% ?
Could you show me how to figure it out..I have been trying but cant think out side the box.

 

[Gary's Student]

Well in A1 thru D8814 I entered:

=RANDBETWEEN(0,200)

and in E1:

=SUM(A1:D1) and copied down

then in another cell:

=COUNTIF(E1:E8814,"<" & 300)/COUNT(E1:E8814)

which yields around .2

 

[JackBean]

With 1,000,000 draws I got 20.3115%.
But a random draw or simulation will only get close to the real probability.

So the Probability Distribution Function (PDF) for each relative is linear,
so is the resulting PDF linear?
No. There is only one way to get 0 or 800,
but more ways to get other totals.

To illustrate how to get a PDF,
take a simpler case (smaller range - integer amounts from 0 to 10)
and with 4 relatives.

Place 4 Columns of numbers 0 to 10 in the starting in Cell A1.
Run the macro:

Dim Kol As Long, Ruw As Long, Rw As Long, Rng As Range
Sub Perm()
Dim Rzt As Variant
Rw = 0
Set Rng = Range("A1").CurrentRegion
Kol = Rng.Columns.Count
Ruw = Rng.Rows.Count
Columns(Kol + 2).ClearContents
ReDim Rzt(1 To Kol)
Perm1 Rzt, 1
End Sub
Sub Perm1(ByVal Rzt As Variant, ByVal Kul As Long)
Dim X As Long, Y As Long
For X = 1 To Ruw
    If Rng(X, Kul) = "" Then Exit Sub
    Rzt(Kul) = Rng(X, Kul)
    If Kul = Kol Then
        Rw = Rw + 1
        For Y = 1 To Kol
            Cells(Rw, Kol + Y + 1) = Rzt(Y)
        Next Y
    Else
        Perm1 Rzt, Kul + 1
    End If
Next X
End Sub

Result:
The 1st 4 Columns on the left are the list.
The 2nd 4 Columns are the first 41 Rows of 11^4=14641 Rows.
The next Column to the right is the first 41 Rows the sum of the 4 Columns.
The 2nd Column from the right is the total of each Count of 0 to 40.
14641 is also the sum of all the Counts in that Column.
The far right Column is the PDF in %,
which is the 2nd Column from the right divided by 14641x100.
[TABLE="width: 519"]
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[TD][/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]14641
[/TD]
[TD="align: right"]1[/TD]
[TD="align: right"]0.0068[/TD]
[/TR]
[TR]
[TD="align: right"]1[/TD]
[TD="align: right"]1[/TD]
[TD="align: right"]1[/TD]
[TD="align: right"]1[/TD]
[TD][/TD]
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[TD="align: right"]1[/TD]
[TD][/TD]
[TD="align: right"]1[/TD]
[TD][/TD]
[TD="align: right"]4[/TD]
[TD="align: right"]0.0273[/TD]
[/TR]
[TR]
[TD="align: right"]2[/TD]
[TD="align: right"]2[/TD]
[TD="align: right"]2[/TD]
[TD="align: right"]2[/TD]
[TD][/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]0[/TD]
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[TD][/TD]
[TD="align: right"]2[/TD]
[TD][/TD]
[TD="align: right"]10[/TD]
[TD="align: right"]0.0683[/TD]
[/TR]
[TR]
[TD="align: right"]3[/TD]
[TD="align: right"]3[/TD]
[TD="align: right"]3[/TD]
[TD="align: right"]3[/TD]
[TD][/TD]
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[TD][/TD]
[TD="align: right"]3[/TD]
[TD][/TD]
[TD="align: right"]20[/TD]
[TD="align: right"]0.1366[/TD]
[/TR]
[TR]
[TD="align: right"]4[/TD]
[TD="align: right"]4[/TD]
[TD="align: right"]4[/TD]
[TD="align: right"]4[/TD]
[TD][/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]4[/TD]
[TD][/TD]
[TD="align: right"]4[/TD]
[TD][/TD]
[TD="align: right"]35[/TD]
[TD="align: right"]0.2391[/TD]
[/TR]
[TR]
[TD="align: right"]5[/TD]
[TD="align: right"]5[/TD]
[TD="align: right"]5[/TD]
[TD="align: right"]5[/TD]
[TD][/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]5[/TD]
[TD][/TD]
[TD="align: right"]5[/TD]
[TD][/TD]
[TD="align: right"]56[/TD]
[TD="align: right"]0.3825[/TD]
[/TR]
[TR]
[TD="align: right"]6[/TD]
[TD="align: right"]6[/TD]
[TD="align: right"]6[/TD]
[TD="align: right"]6[/TD]
[TD][/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]6[/TD]
[TD][/TD]
[TD="align: right"]6[/TD]
[TD][/TD]
[TD="align: right"]84[/TD]
[TD="align: right"]0.5737[/TD]
[/TR]
[TR]
[TD="align: right"]7[/TD]
[TD="align: right"]7[/TD]
[TD="align: right"]7[/TD]
[TD="align: right"]7[/TD]
[TD][/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]7[/TD]
[TD][/TD]
[TD="align: right"]7[/TD]
[TD][/TD]
[TD="align: right"]120[/TD]
[TD="align: right"]0.8196[/TD]
[/TR]
[TR]
[TD="align: right"]8[/TD]
[TD="align: right"]8[/TD]
[TD="align: right"]8[/TD]
[TD="align: right"]8[/TD]
[TD][/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]8[/TD]
[TD][/TD]
[TD="align: right"]8[/TD]
[TD][/TD]
[TD="align: right"]165[/TD]
[TD="align: right"]1.127[/TD]
[/TR]
[TR]
[TD="align: right"]9[/TD]
[TD="align: right"]9[/TD]
[TD="align: right"]9[/TD]
[TD="align: right"]9[/TD]
[TD][/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]9[/TD]
[TD][/TD]
[TD="align: right"]9[/TD]
[TD][/TD]
[TD="align: right"]220[/TD]
[TD="align: right"]1.5026[/TD]
[/TR]
[TR]
[TD="align: right"]10[/TD]
[TD="align: right"]10[/TD]
[TD="align: right"]10[/TD]
[TD="align: right"]10[/TD]
[TD][/TD]
[TD="align: right"]0[/TD]
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[TD="align: right"]0[/TD]
[TD="align: right"]10[/TD]
[TD][/TD]
[TD="align: right"]10[/TD]
[TD][/TD]
[TD="align: right"]286[/TD]
[TD="align: right"]1.9534[/TD]
[/TR]
[TR]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD="align: right"]0[/TD]
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[TD="align: right"]0[/TD]
[TD][/TD]
[TD="align: right"]1[/TD]
[TD][/TD]
[TD="align: right"]360[/TD]
[TD="align: right"]2.4588[/TD]
[/TR]
[TR]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]1[/TD]
[TD="align: right"]1[/TD]
[TD][/TD]
[TD="align: right"]2[/TD]
[TD][/TD]
[TD="align: right"]439[/TD]
[TD="align: right"]2.9984[/TD]
[/TR]
[TR]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]1[/TD]
[TD="align: right"]2[/TD]
[TD][/TD]
[TD="align: right"]3[/TD]
[TD][/TD]
[TD="align: right"]520[/TD]
[TD="align: right"]3.5517[/TD]
[/TR]
[TR]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]1[/TD]
[TD="align: right"]3[/TD]
[TD][/TD]
[TD="align: right"]4[/TD]
[TD][/TD]
[TD="align: right"]600[/TD]
[TD="align: right"]4.0981[/TD]
[/TR]
[TR]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]1[/TD]
[TD="align: right"]4[/TD]
[TD][/TD]
[TD="align: right"]5[/TD]
[TD][/TD]
[TD="align: right"]676[/TD]
[TD="align: right"]4.6172[/TD]
[/TR]
[TR]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]1[/TD]
[TD="align: right"]5[/TD]
[TD][/TD]
[TD="align: right"]6[/TD]
[TD][/TD]
[TD="align: right"]745[/TD]
[TD="align: right"]5.0885[/TD]
[/TR]
[TR]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]1[/TD]
[TD="align: right"]6[/TD]
[TD][/TD]
[TD="align: right"]7[/TD]
[TD][/TD]
[TD="align: right"]804[/TD]
[TD="align: right"]5.4914[/TD]
[/TR]
[TR]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]1[/TD]
[TD="align: right"]7[/TD]
[TD][/TD]
[TD="align: right"]8[/TD]
[TD][/TD]
[TD="align: right"]850[/TD]
[TD="align: right"]5.8056[/TD]
[/TR]
[TR]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]1[/TD]
[TD="align: right"]8[/TD]
[TD][/TD]
[TD="align: right"]9[/TD]
[TD][/TD]
[TD="align: right"]880[/TD]
[TD="align: right"]6.0105[/TD]
[/TR]
[TR]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]1[/TD]
[TD="align: right"]9[/TD]
[TD][/TD]
[TD="align: right"]10[/TD]
[TD][/TD]
[TD="align: right"]891[/TD]
[TD="align: right"]6.0856[/TD]
[/TR]
[TR]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]1[/TD]
[TD="align: right"]10[/TD]
[TD][/TD]
[TD="align: right"]11[/TD]
[TD][/TD]
[TD="align: right"]880[/TD]
[TD="align: right"]6.0105[/TD]
[/TR]
[TR]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]2[/TD]
[TD="align: right"]0[/TD]
[TD][/TD]
[TD="align: right"]2[/TD]
[TD][/TD]
[TD="align: right"]850[/TD]
[TD="align: right"]5.8056[/TD]
[/TR]
[TR]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]2[/TD]
[TD="align: right"]1[/TD]
[TD][/TD]
[TD="align: right"]3[/TD]
[TD][/TD]
[TD="align: right"]804[/TD]
[TD="align: right"]5.4914[/TD]
[/TR]
[TR]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]2[/TD]
[TD="align: right"]2[/TD]
[TD][/TD]
[TD="align: right"]4[/TD]
[TD][/TD]
[TD="align: right"]745[/TD]
[TD="align: right"]5.0885[/TD]
[/TR]
[TR]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]2[/TD]
[TD="align: right"]3[/TD]
[TD][/TD]
[TD="align: right"]5[/TD]
[TD][/TD]
[TD="align: right"]676[/TD]
[TD="align: right"]4.6172[/TD]
[/TR]
[TR]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]2[/TD]
[TD="align: right"]4[/TD]
[TD][/TD]
[TD="align: right"]6[/TD]
[TD][/TD]
[TD="align: right"]600[/TD]
[TD="align: right"]4.0981[/TD]
[/TR]
[TR]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]2[/TD]
[TD="align: right"]5[/TD]
[TD][/TD]
[TD="align: right"]7[/TD]
[TD][/TD]
[TD="align: right"]520[/TD]
[TD="align: right"]3.5517[/TD]
[/TR]
[TR]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]2[/TD]
[TD="align: right"]6[/TD]
[TD][/TD]
[TD="align: right"]8[/TD]
[TD][/TD]
[TD="align: right"]439[/TD]
[TD="align: right"]2.9984[/TD]
[/TR]
[TR]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]2[/TD]
[TD="align: right"]7[/TD]
[TD][/TD]
[TD="align: right"]9[/TD]
[TD][/TD]
[TD="align: right"]360[/TD]
[TD="align: right"]2.4588[/TD]
[/TR]
[TR]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]2[/TD]
[TD="align: right"]8[/TD]
[TD][/TD]
[TD="align: right"]10[/TD]
[TD][/TD]
[TD="align: right"]286[/TD]
[TD="align: right"]1.9534[/TD]
[/TR]
[TR]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]2[/TD]
[TD="align: right"]9[/TD]
[TD][/TD]
[TD="align: right"]11[/TD]
[TD][/TD]
[TD="align: right"]220[/TD]
[TD="align: right"]1.5026[/TD]
[/TR]
[TR]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]2[/TD]
[TD="align: right"]10[/TD]
[TD][/TD]
[TD="align: right"]12[/TD]
[TD][/TD]
[TD="align: right"]165[/TD]
[TD="align: right"]1.127[/TD]
[/TR]
[TR]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]3[/TD]
[TD="align: right"]0[/TD]
[TD][/TD]
[TD="align: right"]3[/TD]
[TD][/TD]
[TD="align: right"]120[/TD]
[TD="align: right"]0.8196[/TD]
[/TR]
[TR]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]3[/TD]
[TD="align: right"]1[/TD]
[TD][/TD]
[TD="align: right"]4[/TD]
[TD][/TD]
[TD="align: right"]84[/TD]
[TD="align: right"]0.5737[/TD]
[/TR]
[TR]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]3[/TD]
[TD="align: right"]2[/TD]
[TD][/TD]
[TD="align: right"]5[/TD]
[TD][/TD]
[TD="align: right"]56[/TD]
[TD="align: right"]0.3825[/TD]
[/TR]
[TR]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]3[/TD]
[TD="align: right"]3[/TD]
[TD][/TD]
[TD="align: right"]6[/TD]
[TD][/TD]
[TD="align: right"]35[/TD]
[TD="align: right"]0.2391[/TD]
[/TR]
[TR]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]3[/TD]
[TD="align: right"]4[/TD]
[TD][/TD]
[TD="align: right"]7[/TD]
[TD][/TD]
[TD="align: right"]20[/TD]
[TD="align: right"]0.1366[/TD]
[/TR]
[TR]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]3[/TD]
[TD="align: right"]5[/TD]
[TD][/TD]
[TD="align: right"]8[/TD]
[TD][/TD]
[TD="align: right"]10[/TD]
[TD="align: right"]0.0683[/TD]
[/TR]
[TR]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]3[/TD]
[TD="align: right"]6[/TD]
[TD][/TD]
[TD="align: right"]9[/TD]
[TD][/TD]
[TD="align: right"]4[/TD]
[TD="align: right"]0.0273[/TD]
[/TR]
[TR]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD][/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]0[/TD]
[TD="align: right"]3[/TD]
[TD="align: right"]7[/TD]
[TD][/TD]
[TD="align: right"]10[/TD]
[TD][/TD]
[TD="align: right"]1[/TD]
[TD="align: right"]0.0068[/TD]
[/TR]
</tbody>[/TABLE]
So for example to get the probability that the sum of all 4 will be no more than 10,
the sum of the first 11 Rows of the Column on the right is 6.83696%.

This approach of listing all the combinations will not work for your example,
because there are too many combinations, with 4 Columns of 0 to 200.

This does not look like a real world situation,
more like homework.

Cat's out of the bag, but we left you something to solve.
Can you come up with the PDF?

By the way you should not post the same question twice,
in two different threads.

 

[TheVillageIdiot]

Brute force iteration method. A more flexible approach would use recursion.

Function ProbQuestion(lower_num As Long, upper_num As Long, target_thresh As Long)
Dim a As Long, b As Long, c As Long, d As Long
Dim tot As Double, cnt As Long, sz As Double

sz = (upper_num - lower_num + 1) ^ 4
For a = lower_num To upper_num
    For b = lower_num To upper_num
        If a + b > target_thresh Then
            Exit For
        Else
            For c = lower_num To upper_num
                If a + b + c > target_thresh Then
                    Exit For
                Else
                    For d = lower_num To upper_num
                        tot = a + b + c + d
                        If tot <= target_thresh Then
                            cnt = cnt + 1
                        Else
                            Exit For
                        End If
                    Next d
                End If
            Next c
        End If
    Next b
Next a
ProbQuestion = cnt / sz
End Function

Assume 0 in A1, 200 in A2, 300 in A3. Call the function as =ProbQuestion(A1,A2,A3)

Change the function name to be more descriptive if necessary

WARNING: This is very slow.

 

[Greg Hitchon]

One way of looking at it is assuming a Normal Distribution (justified with n being so high) and then using some basic properties of this assumed distribution to estimate the probability. Can be done with something like this:

Excel Workbook
	K	L
26	Mean	400
27	St. Dev	116.046
28	P(X	0.194419
Sheet2

Where the standard deviation is calculated from the variance of each random variable (which takes on the values 0-200) and the mean is just the expected value of each relatives gift times 4.

 

[shg]

I did this by convolving the PDFs (which I think is correct) and got 0.202909262

It seems that there should be a way to do this with BINOMDIST.

 

[pgc01]

I followed another (lazier) path, considered this is equivalent of adding several uniform [0,1] uniform distribution.
This makes it a Irwin–Hall distribution, with n=4 and x =1.5 (300/200). I was able to use the known coefficients, in

Irwin

I got a little less: 20.0521%

The quick test I did in the worksheet was:

- 1 million rows with =rand() in columns AD. In column E 1 if the sum of AD is <=1.5, 0 otherwise.

The approximate probability is =Sum(E:E)/1000000

I recalculated several times and the average was very close to the value above.

 

[shg]

Nice, pgc.

Using the CDF at the link, can't that be calculated in closed form as

=1/FACT(4) * COMBIN(4, 0)*(1.5-0)^4 ~ 21.09%