A ball and urn problem

[kristinel]

Here's a practice problem I'm working on today.

There are 25 balls in an urn (14 black and 11 white). We are drawing a sample of five balls.

a) what is the probability that two of the five balls are white?
b) what is the probability that two of the five balls are black?
c) what is the probability that all of the five balls are black?
d) what is the probability that none of the balls are black?

I thought perhaps I needed to use the Poisson distribution formula, but when I looked at my answer, I think I took the wrong approach. What's the proper way to work on this one?

 

[tkhunny]

A tree would be a good place to start, at least to get a feel for it. Of course, this would be the hardest possible way to do it.

First Draw
Pr(Black) = 14/25
Pr(White) = 11/25

Second Draw (After Black)
Pr(Black) = 13/24
Pr(White) = 11/24

Second Draw (After White)
Pr(Black) = 14/24
Pr(White) = 10/24

Third Draw (After Black-Black)
Pr(Black) = ??
Pr(White) = ??

Third Draw (After Black-White)
Pr(Black) = ??
Pr(White) = ??

Third Draw (After White-Black)
Pr(Black) = ??
Pr(White) = ??

Third Draw (After White-White)
Pr(Black) = ??
Pr(White) = ??

 

[kristinel]

That's an interesting approach. I thought about it (using brute force), but I was hoping for another avenue...

Barring an easier avenue, if I were to use a tree diagram, do I just stop after 5 branches of the tree? Given the sample size of my experiment is 5, even though the total number of balls in the urn is 25?

Alternatively, is there a formula I can use that will get me to the answer I want? I always like to have a couple of approaches in my back pocket, it serves me well when I encounter similar problems in the future =-)

 

[pka]

There are 25 balls in an urn (14 black and 11 white).
We are drawing a sample of five balls.
a) what is the probability that two of the five balls are white?
b) what is the probability that two of the five balls are black?
c) what is the probability that all of the five balls are black?
d) what is the probability that none of the balls are black?

This is a simple counting problem.
a) If two are white then three are black: \(\displaystyle \L \frac{ {{11}\choose 2}{{{14}\choose 3}}}{{25}\choose 5}.\)

d) None black, so all white: \(\displaystyle \L \frac{ {{11}\choose 5}}{{25}\choose 5}.\)

 

[kristinel]

Thanks, that's a big help