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 ......?

 

[pgc01]

Hi shg

I believe that, since the value is bigger than 1, you have to add the second term in the summation:

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

which results in the 20.0521%


I did not see this formula yesterday. I'm glad you did, this is a much better way to get the result.

What I did yesterday way to integrate directly the expression in the link for the probability density function:

=(1/6*1^4/4)+1/6*(-3*(1.5^4-1^4)/4+12*(1.5^3-1^3)/3-12*(1.5^2-1^2)/2+4*(1.5-1))

The formula you found is way easier to use.

 

[shg]

My eyes popped open at 4AM and I realized the second term was required ...

Integrating the PDF was clever.

Still, there should be a better way to get the exact answer for the discrete distribution without enumerating the possibilities.

 

[shg]

Here's how I did it for the discrete case, using the simpler example of rolling four six-sided dice numbered 0 to 5 on the faces:

       -A- B C D- -E-
   1   Sum 1 2 3   4 
   2     0 1 1  1   1
   3     1 1 2  3   4
   4     2 1 3  6  10
   5     3 1 4 10  20
   6     4 1 5 15  35
   7     5 1 6 21  56
   8     6   5 25  80
   9     7   4 27 104
  10     8   3 27 125
  11     9   2 25 140
  12    10   1 21 146
  13    11   0 15 140
  14    12   0 10 125
  15    13   0  6 104
  16    14   0  3  80
  17    15   0  1  56
  18    16   0  0  35
  19    17   0  0  20
  20    18   0  0  10
  21    19   0  0   4
  22    20   0  0   1

The array formula in C2:C22 and then copied across is

=CONVOLVE($B$2:$B$22, B2:B22)

The values in B:E are the multiplicity of each sum. If you changed those 1's in col B to 1/6, all the columns would instead show probability.

For the actual problem, there were 800 rows, and I totaled the first 301 (0 to 300).

 

[pgc01]

Hi shg

There is no sign of life from the op, so this is just more out of curiosity.

One piece of info that is missing is what is the smallest amount one can contribute with.

In many denominations (for ex. Euro, Dollar, Pound) we also have cents. In some others maybe even 1/1000's?

From my calculations, 4 cases
- It can be just an integer value, in that case that's your result
- It can have cents
- It can have 1/1000's
- it is a continuous value (theoretical), in this case it's my result

The values I got ( and yours):






<table border="1" cellpadding="1" style="background:#FFF; border-collapse:collapse;border-width:2px;border-color:#CCCCCC;font-family:Arial,Arial; font-size:10pt" ><tr><th style="border-width:1px;border-color:#888888;background:#9CF " > </th><th style="border-width:1px;border-color:#888888;background:#9CF; text-align:center" >A</th><th style="border-width:1px;border-color:#888888;background:#9CF; text-align:center" >B</th><th style="border-width:1px;border-color:#888888;background:#9CF; text-align:center" width=30 >C</th></tr><tr><td style="border-width:1px;border-color:#000000; padding-left:0.5em; padding-top:0.4em; padding-right:0.5em; padding-bottom:0.25em;background:#9CF; text-align:center; " ><b>1</b></td><td style="border-width:1px;border-color:#000000; padding-left:0.5em; padding-top:0.4em; padding-right:0.5em; padding-bottom:0.25em;text-align:right;border-width: 1px;border-color:#888888; "> </td><td style="border-width:1px;border-color:#000000; padding-left:0.5em; padding-top:0.4em; padding-right:0.5em; padding-bottom:0.25em;text-align:right;border-width: 1px;border-color:#888888; "> </td><td style="border-width:1px;border-color:#000000; padding-left:0.5em; padding-top:0.4em; padding-right:0.5em; padding-bottom:0.25em;text-align:right;border-width: 1px;border-color:#888888; "> </td></tr><tr><td style="border-width:1px;border-color:#000000; padding-left:0.5em; padding-top:0.4em; padding-right:0.5em; padding-bottom:0.25em;background:#9CF; text-align:center; " ><b>2</b></td><td style="border-width:1px;border-color:#000000; padding-left:0.5em; padding-top:0.4em; padding-right:0.5em; padding-bottom:0.25em;text-align:left;border-width: 1px;border-color:#888888; ">Integer</td><td style="border-width:1px;border-color:#000000; padding-left:0.5em; padding-top:0.4em; padding-right:0.5em; padding-bottom:0.25em;text-align:right;border-width: 1px;border-color:#888888; ">20.2909%</td><td style="border-width:1px;border-color:#000000; padding-left:0.5em; padding-top:0.4em; padding-right:0.5em; padding-bottom:0.25em;text-align:right;border-width: 1px;border-color:#888888; "> </td></tr><tr><td style="border-width:1px;border-color:#000000; padding-left:0.5em; padding-top:0.4em; padding-right:0.5em; padding-bottom:0.25em;background:#9CF; text-align:center; " ><b>3</b></td><td style="border-width:1px;border-color:#000000; padding-left:0.5em; padding-top:0.4em; padding-right:0.5em; padding-bottom:0.25em;text-align:left;border-width: 1px;border-color:#888888; ">1/100</td><td style="border-width:1px;border-color:#000000; padding-left:0.5em; padding-top:0.4em; padding-right:0.5em; padding-bottom:0.25em;text-align:right;border-width: 1px;border-color:#888888; ">20.0545%</td><td style="border-width:1px;border-color:#000000; padding-left:0.5em; padding-top:0.4em; padding-right:0.5em; padding-bottom:0.25em;text-align:right;border-width: 1px;border-color:#888888; "> </td></tr><tr><td style="border-width:1px;border-color:#000000; padding-left:0.5em; padding-top:0.4em; padding-right:0.5em; padding-bottom:0.25em;background:#9CF; text-align:center; " ><b>4</b></td><td style="border-width:1px;border-color:#000000; padding-left:0.5em; padding-top:0.4em; padding-right:0.5em; padding-bottom:0.25em;text-align:left;border-width: 1px;border-color:#888888; ">1/1000</td><td style="border-width:1px;border-color:#000000; padding-left:0.5em; padding-top:0.4em; padding-right:0.5em; padding-bottom:0.25em;text-align:right;border-width: 1px;border-color:#888888; ">20.0523%</td><td style="border-width:1px;border-color:#000000; padding-left:0.5em; padding-top:0.4em; padding-right:0.5em; padding-bottom:0.25em;text-align:right;border-width: 1px;border-color:#888888; "> </td></tr><tr><td style="border-width:1px;border-color:#000000; padding-left:0.5em; padding-top:0.4em; padding-right:0.5em; padding-bottom:0.25em;background:#9CF; text-align:center; " ><b>5</b></td><td style="border-width:1px;border-color:#000000; padding-left:0.5em; padding-top:0.4em; padding-right:0.5em; padding-bottom:0.25em;text-align:left;border-width: 1px;border-color:#888888; ">Continuous</td><td style="border-width:1px;border-color:#000000; padding-left:0.5em; padding-top:0.4em; padding-right:0.5em; padding-bottom:0.25em;text-align:right;border-width: 1px;border-color:#888888; ">20.0521%</td><td style="border-width:1px;border-color:#000000; padding-left:0.5em; padding-top:0.4em; padding-right:0.5em; padding-bottom:0.25em;text-align:right;border-width: 1px;border-color:#888888; "> </td></tr><tr><td style="border-width:1px;border-color:#000000; padding-left:0.5em; padding-top:0.4em; padding-right:0.5em; padding-bottom:0.25em;background:#9CF; text-align:center; " ><b>6</b></td><td style="border-width:1px;border-color:#000000; padding-left:0.5em; padding-top:0.4em; padding-right:0.5em; padding-bottom:0.25em;text-align:right;border-width: 1px;border-color:#888888; "> </td><td style="border-width:1px;border-color:#000000; padding-left:0.5em; padding-top:0.4em; padding-right:0.5em; padding-bottom:0.25em;text-align:right;border-width: 1px;border-color:#888888; "> </td><td style="border-width:1px;border-color:#000000; padding-left:0.5em; padding-top:0.4em; padding-right:0.5em; padding-bottom:0.25em;text-align:right;border-width: 1px;border-color:#888888; "> </td></tr><tr><td colspan=4 style="background:#9CF; padding-left:1em" > [Book1]Sheet0</td></tr></table>


I will come back to this when I have the time, I'd really like to understand your convolution solution. Statistics is, however, not my area of expertise and I'll have to do some digging to get it.

 

[shg]

A very good summary, pgc, and an acute observation about less than whole-dollor donations. I would add that the sum of uniform distributions converges quickly to a normal distribution, so Greg's solution is also a very good approximation.

My convolution code is at http://www.mrexcel.com/forum/excel-...e-convolutions-visual-basic-applications.html.

When looking for recurrence relations, the OEIS is my first stop. The sequence for the multiplicities of rolls of four dice, for example, is A063260 - OEIS, and the formula given,

a(n, k)= sum(a(n-1, k-j), j=0..5) else.

is a convolution.

Statistics is, however, not my area of expertise

Nor mine, but it fascinates me, and so motivates me to struggle.

 

[shg]

Clarification -- the sequence is for the multiplicities of sums for rolls of any number of dice, starting with rolling none:

1,
1, 1, 1, 1, 1, 1,
1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1,
1, 3, 6, 10, 15, 21, 25, 27, 27, 25, 21, 15, 10, 6, 3, 1,
1, 4, 10, 20, 35, 56, 80, 104, 125, 140, 146, 140, 125, 104, 80, 56, 35, 20, 10, 4, 1,
...

 

[shg]

@pgc: I do realize that your examples are exact solutions in the progression of successively finer discrete cases, where Greg's solution is the limit of increasing the number of donors, and so not in the same family. What I meant was that his was a good observation of another aspect.

And thanks for the intro to the Irwin distribution.

(Sorry for the monologue.)

 

[pgc01]

shg, thank you for the links. I will surely study them.

 

[Jagdeepthomas]

If all the relatives will give you 75$, then this is equal probability 1:1:1:1 if only three relatives provide you with money then each relative have to offer you with 100$ each and so on.