# [SOLVED] Choice and chance

• Aug 15th 2007, 02:49 AM
dcwmorley
[SOLVED] Choice and chance
Please could someone recommend me a book which might help me understand the following problem :

"A swimming pool is emptied and then filled with several hundred thousand small plastic balls. There are black balls and there are white balls. The fraction of the balls that are black is x. I then throw in a small bucket and extract twelve balls. What is the chance that at least one ball in the bucket is a black one ?"

I might have to ask myself other similar questions.

I have a few algebra texts which deal with similar but seemingly less complicated questions of the same general type, but I'm not really sure what sort of book I'm looking for.

David Morley
• Aug 15th 2007, 03:54 AM
dodoldotkom
maybe you could try something like "introduction to probability". Or check this site Untitled Document
• Aug 15th 2007, 03:59 AM
JakeD
Quote:

Originally Posted by dcwmorley
Please could someone recommend me a book which might help me understand the following problem :

"A swimming pool is emptied and then filled with several hundred thousand small plastic balls. There are black balls and there are white balls. The fraction of the balls that are black is x. I then throw in a small bucket and extract twelve balls. What is the chance that at least one ball in the bucket is a black one ?"

I might have to ask myself other similar questions.

I have a few algebra texts which deal with similar but seemingly less complicated questions of the same general type, but I'm not really sure what sort of book I'm looking for.

David Morley

You are looking for a book on probability. Here is a free online text that may work for you.

http://www.dartmouth.edu/~chance/tea.../booktitle.gif
• Aug 16th 2007, 03:36 PM
Soroban
Hello, David!

Quote:

A swimming pool is emptied and then filled with several hundred thousand small plastic balls.
There are black balls and there are white balls.
The fraction of the balls that are black is $x$.
I then throw in a small bucket and extract twelve balls.
What is the chance that at least one ball in the bucket is a black one ?

They don't tell us the exact number of balls in the swimming pool,
. . but say there are "several hundred thousand" of them.

I must assume that, for drawing a small sample (a dozen balls),
. . the probability of getting a black ball is always $x$.
Then this is a Binomial probability.

We have: . $P(black) = x,\;\;P(white) = 1-x$

The opposite of "at least one black ball" is "all white balls"
. . and: . $P(\text{all white}) \:=\:(1-x)^{12}$

Therefore: . $P(\text{at least one black}) \:=\:1 - (1-x)^{12}$