Expert: Paul Klarreich - 2/15/2009

Question
If you could help me answer this question i would be much obliged. I have already answered parts a,b,c but i cant answer (d) and (e)

An Urn contains 15 balls, numbered 1 to 15. The balls in the urn are thoroughly mixed and three balls are selected at random from the urn in such a way that any subset of three balls are equally likely to be selected. The sample space for this experiment can be described as
S = {{i,j,k} : 1≤i,j,k≤15 and i≠j,j≠k,i≠k}

(d) find the probability that at least one even numbered ball and at least one odd numbered ball are selected.

(e) Find the conditional probability that the all the numbers on the balls are divisible by '3', given that at least one of the numbers is divisible by'3'

It would be really great If you could help me to answer these questions.

Kind Regards,
Mike

Answer
Questioner:   mike88
Category:  Advanced Math
Private:  Yes
 
Subject:  Probabilities
Question:  If you could help me answer this question i would be much obliged. I have already answered parts a,b,c but i cant answer (d) and (e)

An Urn contains 15 balls, numbered 1 to 15. The balls in the urn are thoroughly mixed and three balls are selected at random from the urn in such a way that any subset of three balls are equally likely to be selected. The sample space for this experiment can be described as
S = {{i,j,k} : 1 <= i, j , k <= 15 and i/=j,j/=k,i/=k}

(d) find the probability that at least one even numbered ball and at least one odd numbered ball are selected.

(e) Find the conditional probability that the all the numbers on the balls are divisible by '3', given that at least one of the numbers is divisible by'3'

It would be really great If you could help me to answer these questions.

Kind Regards,
Mike
 
........................................
For a,b,c I assume you already found   C(15,3) as the number of 3-ball subsets.

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For all problems of this type (What type, you ask?  I mean problems that ask about 'at least one something or other')  I recommend that you look at their opposites, as follows:

Dor (d), Try this approach:

How many subsets contain ONLY odds?   C(8,3)
How many subsets contain ONLY evens?  C(7,3)

How many others are there?  C(15,3) - (C(8,3) + C(7,3)) --- this will be the number of subsets with at least one of each.


(e) Find the conditional probability that the all the numbers on the balls are divisible by '3', given that at least one of the numbers is divisible by'3'

How many subsets contain at least one 'divisible by 3'?  Better you should ask:

How many do not?  C(10,3)

So how many do?  C(15,3) - C(10,3)   << your denominator.

Now how many contain ONLY 'divisible by 3'?   C(5,3)

That should do it.

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Notes:

1. Don't use special symbols at this site.  They just screw things up.  Write  <=, >=, and /= (not equal).  I changed them, as you can see.

2. Don't mark PRIVATE.  I change it, anyway.  If you don't want anyone to see your question or answer, you have to send it elsewhere.