HSC 2008 Question 7 a) (1 Viewer)

[roryclifford]

Hey guys,

I don't know how to do this probability question from 2008, could someone please help me with it? I can get out the 1st part (though only sort of understand) but the rest is lost to me.

An urn contains n red balls, n white balls and n blue balls. Three balls are drawn at random from the urn, one at a time, without replacement.
(i) What is the probability, ps, that the three balls are all the same colour?
(ii) What is the probability, pd, that the three balls are all of different colours?
(iii) What is the probability, pm, that two balls are of one colour and the third
is of a different colour?
(iv) If n is large, what is the approximate ratio ps: pd : pm?

http://www.boardofstudies.nsw.edu.a...s/pdf_doc/2008HSC-mathematics-extension-2.pdf

Thanks

 

[math man]

There are in total 3n balls.

For i) The probability of getting the same colour when 3 balls are drawn is as follows(noting there is no replacement):



Note it will be the same probability if it were red, blue or white so we just times the probability of getting on colour 3 times by 3. This simplifies to:



part i is done.

For part ii) we want different colours, which can occur as follows:



as there are 6 different possibilites for different colours:

RGB RBG BGR BRG GRB GBR and the probabilities will be the same each time.



You see here that the number of red, blue and green dont change, only the number of balls in the urn.



and part ii is done.

Now for part iii you can do it 2 ways, first you can work out all the possibilities you can get for 2 colours and one different colour, or you can realise that:



as there only combinations of 3 colours can be all same, all different or 2 same one diff.

Therefore:



Simplifying this becomes:



Expanding this becomes:



which we simply further to:



and part iii is done.

For part iv by n being large we take the limit as n approaches infinity for each and compare the ratios.

For i) :



For ii) :



For iii) :



Therefore:



for large n

 

[roryclifford]

There are in total 3n balls.

For i) The probability of getting the same colour when 3 balls are drawn is as follows(noting there is no replacement):



Note it will be the same probability if it were red, blue or white so we just times the probability of getting on colour 3 times by 3. This simplifies to:



part i is done.

For part ii) we want different colours, which can occur as follows:



as there are 6 different possibilites for different colours:

RGB RBG BGR BRG GRB GBR and the probabilities will be the same each time.



You see here that the number of red, blue and green dont change, only the number of balls in the urn.



and part ii is done.

Now for part iii you can do it 2 ways, first you can work out all the possibilities you can get for 2 colours and one different colour, or you can realise that:



as there only combinations of 3 colours can be all same, all different or 2 same one diff.

Therefore:



Simplifying this becomes:



Expanding this becomes:



which we simply further to:



and part iii is done.

For part iv by n being large we take the limit as n approaches infinity for each and compare the ratios.

For i) :



For ii) :



For iii) :



Therefore:



for large n

Ah sweet, thanks, explained it perfectly. You really are the math man!