Combinatorial Probabilities Without Replacement

[Amateurhr]

Let's say I have 40 balls in this distribution:
Type A: 25
Type B: 10
Type C: 5


If I am going to blindly pull out 10 balls without replacement, what are the chances I get AT LEAST:
5 of Type A
3 of Type B


Any ideas how to create this equation and if Excel has any formulas that make this simpler? I need an equation because I have about 100+ of these calculations to perform. If not, any idea how to create a monte carlo simulation of this?


Thanks in advance folks!


-Tommy

 

[West Man]

5 of Type A
3 of Type B

Are these 2 separate questions or are you looking for the probability of AT LEAST 5 A AND AT LEAST 3 B. if the latter it is much more complex

 

[J.Ty.]

Each row corresponds to a single choice of the quantities of the balls of each color, so that they satisfy your criteria.
Column D is the calculation of the probability of this particular combination of the quantities occurring.
The total probability is the sum over all rows.

J.Ty.

Excel 2010
	A	B	C	D
1	A	B	C	Probability
2	5	3	2	0,075214
3	5	4	1	0,065812
4	5	5	0	0,015795
5	6	3	1	0,125357
6	7	3	0	0,068051
7	6	4	0	0,043875
8			Total prob.	0,394104
Sheet1

Cell Formulas
Range	Formula
D2	=(COMBIN(25,A2)*COMBIN(10,B2)*COMBIN(5,C2))/COMBIN(40,10)
D3	=(COMBIN(25,A3)*COMBIN(10,B3)*COMBIN(5,C3))/COMBIN(40,10)
D4	=(COMBIN(25,A4)*COMBIN(10,B4)*COMBIN(5,C4))/COMBIN(40,10)
D5	=(COMBIN(25,A5)*COMBIN(10,B5)*COMBIN(5,C5))/COMBIN(40,10)
D6	=(COMBIN(25,A6)*COMBIN(10,B6)*COMBIN(5,C6))/COMBIN(40,10)
D7	=(COMBIN(25,A7)*COMBIN(10,B7)*COMBIN(5,C7))/COMBIN(40,10)
D8	=SUM(D2:D7)

 

[Amateurhr]

West Man, I am looking for AT LEAST 5 A AND AT LEAST 3 B

Looks like J Ty has created an answer but the issue is, I think I would have to write a macro to accomplish that since I have 100 of these cells to do and each seems to require that I enumerate every possible outcome (which may be 6 in this case), which changes for every combination of balls and requirements.

If that's the only possible way, then I don't mind going with that. Any idea how the formulas might change as a macro?

 

[J.Ty.]

I did it for the requirement "at least 5 A and and least 3 B".
As far as I understand, you have 99 other pairs (triples?) of requirements.
Can you post a representative sample?

J.Ty.