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Balls in Boxes, Probability Question

by dogma
Tags: balls, boxes, probability
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dogma
#1
Dec3-04, 05:44 AM
P: 35
Hello one and all. I could use a little guidance here on a probability problem.

Box #1 contains a black balls and b white balls while box #2 contains c black balls and d white balls. A ball is chosen randomly from box #1 and placed in box #2. A ball is then randomly chosen from box #2 and placed in box #1. What is the probability that box #1 still has a black balls and b white balls?

Okay, from that I come up with the following:

Let Random Variable X1 = a black ball is transferred to box #2 from box #1
Let R.V. X2 = a white ball is transferred to box #2 from box #1

Let R.V. Y1 = a black ball is transferred to box #1 from box #2
Let R.V. Y2 = a white ball is transferred to box #1 from box #2

[tex]P(X_1) = \frac{a}{a+b}[/tex] and [tex]P(X_2) = \frac{b}{a+b}[/tex]

[tex]P(Y_1 \mid X_1) = \frac{c+1}{c+d+1}[/tex] and [tex]P(Y_1 \mid X_2) = \frac{c}{c+d+1}[/tex]

[tex]P(Y_2 \mid X_1) = \frac{d}{c+d+1}[/tex] and [tex]P(Y_2 \mid X_2) = \frac{d+1}{c+d+1}[/tex]

This is where I get stuck.

I know (for example) that I can define another R.V. to represent, say, a black ball was selected from box #2 (I'll call it R.V. A), and...

[tex]P(A) = P(X_1) \cdot P(Y_1 \mid X_1) + P(X_2) \cdot P(Y_1 \mid X_2)[/tex]

Assuming I'm somewhat on the right track and haven't screwed things up, how would I go about determining the probability that box #1 still has a black balls and b white balls?

Thanks in advance for your enlightenment (and do I need it).

dogma
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matt grime
#2
Dec3-04, 06:18 AM
Sci Advisor
HW Helper
P: 9,398
You're on the right track by thinking conditionally.


Can I make some notational changes and abuses?

P(correct allocation of balls) = P(correct number given a white transferred or correct n umber given a black transferred)

mutuall exclusive

=P(correct number given white transferred) +P(correct number given black transferred)

=P(white put back given white taken) + P(balck put back given black taken)

= somethings you've worked out.
dogma
#3
Dec3-04, 09:36 AM
P: 35
First of all, thanks for your response. I greatly appreciate your help.

My mind is swimming…so hopefully I'm not going to make this worse. If I'm correctly utilizing the info you provided:

P(correct allocation of balls) = P(a black balls and b white balls), since box #1 started off with a black balls and b white balls [and box #2 started off with c black balls and d white balls].

P(white ball chosen from box #1) =[tex]\frac{b}{a+b}[/tex]

P(black ball chosen from box #1) =[tex]\frac{a}{a+b}[/tex]

and

P(white ball chosen from box #2 given a white ball chosen from box #1) =[tex]\frac{d+1}{c+d+1}[/tex]

P(black ball chosen from box #2 given a black ball chosen from box #1) =[tex]\frac{c+1}{c+d+1}[/tex]

and finally,the correct allocation of balls:

P(a black balls and b white balls) = [tex]\frac{b}{a+b} \cdot \frac{d+1}{c+d+1}+\frac{a}{a+b} \cdot \frac{c+1}{c+d+1}[/tex]

I'm still a little fuzzy about this…then again, using Playdoh is challenging for me.

Am I warmer, colder, or way out in left field?

Thanks again!

dogma

matt grime
#4
Dec3-04, 09:58 AM
Sci Advisor
HW Helper
P: 9,398
Balls in Boxes, Probability Question

yp, that seems to be about right (and correcting any errors i may have made).
dogma
#5
Dec3-04, 10:39 AM
P: 35
thank you for your guidance and wisdom.

best of wishes,

dogma
dogma
#6
Dec3-04, 01:42 PM
P: 35
Follow on question:

Could I have used a hypergeometric distribution to figure this out? I guess I would just have to figure out how to set it up that way.

dogma
Lele
#7
Dec3-04, 11:40 PM
P: 4
One can use the hypergeometic distribution to get to the same result. Using the conditional probability approach, the probabilities (for choosing the black or white ball from box#1 )and the conditional probabilities (of choosing the black or white balls from box#2) are the same as the one you got earlier.

Lets see for one case:
Box#1: choosing a black ball
Probability = (a choose 1) * (b choose 0) / (a+b choose 1)
= a/(a+b)
(a choose 1 : ways of choosing 1 ball from a black balls)
Similarly the probability of choosing a black ball from box# given black ball chosen from box#1 = (c+1)/ (c+d+1)

This is exactly the first term in your equation.

I would appreciate if someone could let me know how to handle mathematical notation in this text editor.
dogma
#8
Dec4-04, 05:27 AM
P: 35
Quote Quote by Lele
I would appreciate if someone could let me know how to handle mathematical notation in this text editor.
Lele,

Check out this link on LaTex (it's in the General Physics forum): http://physicsforums.com/showthread.php?t=8997

It is a thread containing info about LaTex typesetting in a message. It's pretty easy to do.

Thanks and good luck.

dogma


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