Question

Statistics & Probabilities

Answer 1

An urn contains 40 balls with 10 white, 25 black, 5 red. From this box
are extracted three balls. Determine the probability that the balls of different colors to be extracted.

Answer 2

Well I can definetly see that the total number of cases is:C(3,40) but how do I find out the favorable one's?

Answer 3

This is a super slow method, but I really like it because it's very easy to follow. I essentially make a "tree" of possibilities:

Answer 4

And so on and so forth. You'd have 3 "Branches." So the chance of getting 3 white balls would be, after the third branch, 10/40 x 9/39 x 8/38

You can mark which branches end in what colors and add up results if absolutely necesary

Answer 5

(I don't actually recommend this method in terms ofpracticality, if you're able to figure out the equation/ other faster method that's great, but this is super easy to visualize and good if you're really super stucK)

Answer 6

@Hitaro9 thx as you said your method is applyable only for a very "finite" number of cases but I'm trying to catch the general method.

Answer 7

Yeah definitely. I would bump the question and see if someone more knowledgable with statistics is able to help, but if it starts to get late and you're just tired of waiting, that option is there for ya'.

Answer 8

Good luck, hopefully someone else knows a better way to do this.

Answer 9

So simplifying my thing a bit, if you're only looking for how likely it is to pull a single ball, you can do something like

1- the chance of not getting it 3 times in a row

So like for white you do

1-(30/40)(29/39)(28/38)

And the same thing for the remaining 2 colors

Answer 10

Would that make sense?

Answer 11

http://www.wolframalpha.com/input/?i=%2810*25*5%29%2F%2840+choose+3%29+

Answer 12

my guess is that since all of the balls must be different and the red balls are only 5 than the favorable cases are 5*5*5=125 and the probability is:125/C(3,40)

Answer 13

All the balls must be different
that means you must pick 1 white ball, 1 black ball and 1 red ball.

you can pick 1 white ball from 10 white balls in "10" ways
you can pick 1 black ball from 25 black balls in "25" ways
you can pick 1 red ball from 5 red balls in "5" ways

so number of favorable ways of getting all 3 different balls = 10*25*5

Answer 14

there are two ways I can think of to do this.
the first way is a bit longer, you can write down the events explicitly.
Let W = white, B=black, R = red
WBR
WRB
RBW
RWB
BRW
BWR

THe probability of WBR is
10/40 * 25/39*5/38
THe probability of WRB is
10/40 * 5/39*25/38
... etc

Notice that you have the same numerator and denominator , except rearranged. So the total probability is
6 * 10/40 * 25/39*5/38 = .1265


the faster way to do it is with combinatorial notation.

$$ \LARGE {\frac{ \binom {10}{ 1}\binom {25}{ 1}\binom {5}{ 1}}{\binom {40}{3} }
= .1265

} $$