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A certain jar contains only b black marbles, w white

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A certain jar contains only b black marbles, w white [#permalink] New post 30 Mar 2008, 10:43
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A
B
C
D
E

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A certain jar contains only b black marbles, w white marbles, and r red marbles. If one marble is to be chosen at random from the jar, is the probability that the marble chosen will be red greater than the probability that the marble chosen will be white?

1) r/(b+w) > w/(b+r)

2) b - w > r
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Re: DS marble probability [#permalink] New post 30 Mar 2008, 11:03
young_gun wrote:
A certain jar contains only b black marbles, w white marbles, and r red marbles. If one marble is to be chosen at random from the jar, is the probability that the marble chosen will be red greater than the probability that the marble chosen will be white?

1) r/(b+w) > w/(b+r)

2) b - w > r



I think the correct answer is E
A is insufficient because r/(b+w) does not take into account r in the denominator and w/(b+r) does not include w in the denominator.
B is sufficient because if b-w>r
If b=10 w=2 then 8>r
In this case P (R) can be greater or less than p(W)
C I cannot see a relationship
D Insufficient
E is the correct answer

What is the OA?
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Re: DS marble probability [#permalink] New post 30 Mar 2008, 12:06
I think the OA is A...I'm not sure why though.
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Re: DS marble probability [#permalink] New post 31 Mar 2008, 19:31
1) r/(b+w) > w/(b+r)

because if the ratio of red to the rest is higher than the ratio of white to the rest while blacks remain constant then there must be more red than white

r > w * (b+w) / (b+r)
r(b+r) > w(b+w)
set b = 0
r^2 > w^2
r > w
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Re: DS marble probability [#permalink] New post 01 Apr 2008, 01:22
I choose A as the answer option. Essentially we have to find is...
r/(b+w+r) > w/(b+w+r)

By statement 1
r/(b+w) > w/(b+r) or
r(b+r) > w(b+w) or
rb + r^2 > wb + w^2 or
rb + r^2 + wr > wb + w^2 + wr (adding wr to both sides)
or simply
r/(b+w+r) > w/(b+w+r) (if you group the factors)

Hence the condition is sufficient. (ruling out b, c & d)

By statement 2
b - w > r
or b > r+w which means b is greater than the sum of r+w, but we won;t know the relation between r & w. This statement is not sufficient. Hence A
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Re: DS marble probability [#permalink] New post 01 Apr 2008, 09:16
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r/(b+w) > w/(b+r)
Add 1 to both sides
r/(b+w) +1 > w/(b+r) +1
(r+b+w)/(b+w) > (r+b+w)/(b+r)
1/(b+w) > 1/(b+r)
inverting

(b+w) < (b+r)

w<r hence sufficient
Re: DS marble probability   [#permalink] 01 Apr 2008, 09:16
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