Probability - Choosing balls out of a box.

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Brushes

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Jul 19, 2021
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Hello!
I have encountered the following question, while studying for probability course:

We have a box that includes 9 balls: 4 balls painted black, 3 balls painted white, 2 balls painted red.
We take out 4 balls out of the box, without returning.

There are 2 questions, one of them I've managed to solve, and I would like to hear if I was right or not.
The second one I can't figure the answer.

1. What is the probability that all the four balls are from the same color?
The only option is to pick the 4 balls painted black. So it's (4 4) / (9 4) which is 1/126.


2. What is the probability that every ball will be in different color?
(For example, if we pick black, then white, then red, then black).
I've been trying calculating them case by case, but I ended up with tons of cases to calculate. I'm looking for a faster way.
I've been thinking of going 1-(cases), but couldn't figure out which cases to check.

What supposed to be my state of mind while solving question no.2?
Thanks in advance!
 
Brushes said:
Hello!
I have encountered the following question, while studying for probability course:

We have a box that includes 9 balls: 4 balls painted black, 3 balls painted white, 2 balls painted red.
We take out 4 balls out of the box, without returning.

There are 2 questions, one of them I've managed to solve, and I would like to hear if I was right or not.
The second one I can't figure the answer.

1. What is the probability that all the four balls are from the same color?
The only option is to pick the 4 balls painted black. So it's (4 4) / (9 4) which is 1/126.


2. What is the probability that every ball will be in different color?
(For example, if we pick black, then white, then red, then black).
I've been trying calculating them case by case, but I ended up with tons of cases to calculate. I'm looking for a faster way.
I've been thinking of going 1-(cases), but couldn't figure out which cases to check.

What supposed to be my state of mind while solving question no.2?
Thanks in advance!
Click to expand...
Assuming you still are choosing 4 balls, the answer is trivial. How many ways are there to choose 4 balls, all of different colors, given only 3 colors? (Two black balls would not be a valid choice. I don't think the problem means that each ball is a different color than the one before it. Or did you not quote the exact problem?)

Now suppose you are picking only 3 balls, and all must be different.

There are, as usual, several different ways to approach this; but let's use combinations as you did.

How many ways are there to choose one of each color, without regard to order? We just have to choose 1 of the 4 black, 1 of the 3 white, and 1 of the 2 red. How many ways do you get?
 
@Dr.Peterson
Thank you for your comment.
I understand that if we pick 4 balls, from a box that contains only 3 colors of balls, for sure we pick atleast 2 of the same color.
I still think that the idea behind the question is each ball is a different color than the one before it. Because this question is marked as a hard question by our lecturer.
Can we even do it with basic combination tools?
I thought about writing all the possible combinations, like:
1.black-red-white-red
2.black-red-white-black
and so on. But that would make me calculate tons of possible combinations.
 
So you're saying you quoted the problem exactly, and it is, word for word, this?

We have a box that includes 9 balls: 4 balls painted black, 3 balls painted white, 2 balls painted red.​
We take out 4 balls out of the box, without returning.​
​
1. What is the probability that all the four balls are from the same color?​
2. What is the probability that every ball will be in different color?​

The second question just doesn't mean what you are taking it to mean. And a trivial question can also be difficult, if it tricks the reader!

But let's suppose it really said this:

2'. What is the probability that every ball will be a different color than the ball chosen immediately before it?​

You're right that this would be very complicated. So my next question is, what topic was most recently taught before this problem was assigned? And what other methods have you learned that you might be expected to be able to use? So far, I have no idea what level you are at, except that you know about combinations.
 
Brushes, if you are studying for a probability course, then you are not in the course yet?
If that is the case, you must be using a booklet
or software that includes the final answers?

I would forget about the second question and
move on to others, because this is a needless
drain on you and other math helpers.

If you mean you are in the course with an instructor, please ask that instructor for. a
clarification on the question.
 
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