Closed form expression for this?

AI Thread Summary
The discussion focuses on calculating the probability of drawing n red balls in a row from an urn containing NA blue balls and NB red balls. The initial probability of drawing a red ball is given as NA/(NA+NB), and it adjusts with each successive draw as red balls are removed. The correct formula for the probability of drawing n successive red balls is presented as p(n) = ∏i=0n(NA-1)/(NA+NB-i). Additionally, the probability can be expressed using factorials and binomial coefficients, leading to the closed form p(n) = C(R,n)/C(N,n), where C(a,b) denotes the binomial coefficient. This formulation clarifies the probability calculation for drawing multiple red balls in succession.
aaaa202
Messages
1,144
Reaction score
2
Say we have NA blue balls and NB red balls mixed together in an urn. When we pick a ball it leaves the urn. I want to find the probability of picking n red balls in a row.
The probability of picking a red ball is:
NA/(NA+NB)
But each time a red ball leaves the urn the probability of picking the next will be:
(NA-1)/(NA+NB-1)
So in general for n successive red ball picks the probability must be:

p(n) = ∏i=0n(NA-1)/(NA+NB-i)

First of all I want to ask if this is correctly deduced. Secondly, I want to know if you can evaluate a product like the above in a closed form.
 
Physics news on Phys.org
aaaa202 said:
Say we have NA blue balls and NB red balls mixed together in an urn. When we pick a ball it leaves the urn. I want to find the probability of picking n red balls in a row.
The probability of picking a red ball is:
NA/(NA+NB)
But each time a red ball leaves the urn the probability of picking the next will be:
(NA-1)/(NA+NB-1)
So in general for n successive red ball picks the probability must be:

p(n) = ∏i=0n(NA-1)/(NA+NB-i)

First of all I want to ask if this is correctly deduced. Secondly, I want to know if you can evaluate a product like the above in a closed form.

I'll change to a nicer notation: we start with N balls altogether, R red and B blue, with R+B=N. If you draw n <= R balls, what is the probability all n are red? Note: this is very different from asking that if we pick all balls, one at a time, we ever get n red balls in a row; that would be a much harder problem.

As you said, the answer is
p(n) = \frac{R}{N} \frac{R-1}{N-1} \cdots \frac{R-n+1}{N-n+1}.
This can be written in terms of factorials and binomial coefficients (if you think that is nicer and fits your definition of "closed form"):
p(n) = \frac{R!}{(R-n)!} \frac{(N-n)!}{N!}= \frac{C(R,n)}{C(N,n)},
where C(a,b) = \frac{a!}{b!(a-b)!} is the binomial coefficient.

RGV
 
I picked up this problem from the Schaum's series book titled "College Mathematics" by Ayres/Schmidt. It is a solved problem in the book. But what surprised me was that the solution to this problem was given in one line without any explanation. I could, therefore, not understand how the given one-line solution was reached. The one-line solution in the book says: The equation is ##x \cos{\omega} +y \sin{\omega} - 5 = 0##, ##\omega## being the parameter. From my side, the only thing I could...
Back
Top