Marbles Probability

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ieeyorei: Newbie | Next Rank: 10 Posts; Posts: 8; Joined: Sat Aug 16, 2008 3:46 pm; Thanked: 1 times

by ieeyorei » Wed Mar 11, 2009 7:04 pm

A certain jar contains only b black marbles, w white marble, and r red marble. 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

A [/spoiler]

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scoobydooby: Legendary Member; Posts: 1035; Joined: Wed Aug 27, 2008 10:56 pmFollowed by: 1 member; Thanked: 104 times

by scoobydooby » Wed Mar 11, 2009 9:52 pm

P(r)=r/(r+b+w)
P(w)=w/(r+b+w)

simplifying the stimulus,
Q. P(r)>P(w)?
=>r>w?

1) r/(b + w) > w/(b + r)
=>r/(b + w)-w/(b + r)>0
=>r(b+r)-w(b+w)/{(b+w)(b+r)}>0
=>rb+r^2-wb-w^2>0
=>b(r-w)+r^2-w^2>0
=>b(r-w)+(r+w)(r-w)>0
=>(r-w)(b+r+w)>0
=>(r-w)>0 => r>w sufficient

2) b-w > r
cant say from here whether r>w not sufficient

hence A

=

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Ian Stewart: GMAT Instructor; Posts: 2623; Joined: Mon Jun 02, 2008 3:17 amFollowed by: 355 members; Location: Montreal; Thanked: 1090 times; GMAT Score:780

Re: Marbles Probability

by Ian Stewart » Thu Mar 12, 2009 4:48 am

ieeyorei wrote:A certain jar contains only b black marbles, w white marble, and r red marble. 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

A [/spoiler]

There are quite a few ways to look at this problem, though it's actually based on an idea we encounter all the time in day to day life. I'll suggest two other approaches:

If you know the concept of 'odds' from probability (and from real life), then the first statement is clearly sufficient here. If you say that "the odds are 21 to 5 in favor of picking a consonant when you pick a random letter from the alphabet", that is just a ratio of the number of ways to pick a consonant to the number of ways to not pick a consonant. From those odds, we can see that the probability of picking a consonant is 21/26, and the probability of picking a vowel is 5/26.

That's what statement 1 is telling us: the odds of getting a red marble are better than the odds of getting a white marble (the fraction on the left is the ratio of red marbles to non-red marbles, while the fraction on the right is the ratio of white marbles to non-white marbles). If the odds are higher, the probability must be higher.

Alternatively, you could look at this abstractly. We don't need to rewrite the fractions at all. We know that:

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

and all of the letters represent positive quantities. We also know that if a fraction has *both* a larger numerator and a smaller denominator than another, it must be larger in value. Looking at the above inequality, it's impossible for w to be greater than r; if it were, then the fraction on the right would have both a larger numerator and a smaller denominator than the fraction on the left. So r must be greater than w (they can't be equal, if that inequality is true), which is all we need to know here.

For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

ianstewartgmat.com

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cramya: Legendary Member; Posts: 2467; Joined: Thu Aug 28, 2008 6:14 pmFollowed by: 11 members; Thanked: 331 times

by cramya » Thu Mar 12, 2009 8:08 pm

Simply superb explanation; thanks Ian!

Wish we could make a book out of Stuart's/your explanations for all of us.....

No kidding I still cant believe where u could have missed that 20 in your GMAT.

Regards,
CR

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abcdefg: Master | Next Rank: 500 Posts; Posts: 122; Joined: Mon Sep 22, 2008 5:11 pm; Thanked: 1 times

by abcdefg » Sun Jul 19, 2009 9:22 pm

Ian your solution makes perfect sense. I'm just wondering if you can help me clarify this thought:

If R/W > W/R then we know that R > W. But there's a "b" in the fraction, so it's talking about the ratio of red marbles to non-red marbles and white marbles to non-white marbles. However how does ratio of Red vs. Non-red being higher than white vs. non white show us that red > white?

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Domnu: Master | Next Rank: 500 Posts; Posts: 385; Joined: Sun May 24, 2009 3:55 pm; Thanked: 11 times; GMAT Score:740

by Domnu » Mon Jul 20, 2009 7:13 pm

Here's another mathematical solution...

A -

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

(b+w)/r < (b+r)/w

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

(b+w+r)/r < (b+w+r)/w

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

so P(r) > P(w). Here, we relied on the fact that a > b -> 1/a < 1/b.

Have you wondered how you could have found such a treasure? -T

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gmat1978: Senior | Next Rank: 100 Posts; Posts: 35; Joined: Thu Dec 23, 2010 7:38 pm

by gmat1978 » Wed Jun 08, 2011 11:14 pm

Ian,

I am unable to follow your logic below. Can you please provide any examples?

Thanks

Ian Stewart wrote: Alternatively, you could look at this abstractly. We don't need to rewrite the fractions at all. We know that:

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

and all of the letters represent positive quantities. We also know that if a fraction has *both* a larger numerator and a smaller denominator than another, it must be larger in value. Looking at the above inequality, it's impossible for w to be greater than r; if it were, then the fraction on the right would have both a larger numerator and a smaller denominator than the fraction on the left. So r must be greater than w (they can't be equal, if that inequality is true), which is all we need to know here.

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Frankenstein: Legendary Member; Posts: 1448; Joined: Tue May 17, 2011 9:55 amFollowed by: 53 members; Location: India; Thanked: 375 times

by Frankenstein » Thu Jun 09, 2011 12:35 am

gmat1978 wrote:Ian,

I am unable to follow your logic below. Can you please provide any examples?

Thanks

Hi,
Lets say r>w. So b+r > b+w
Consider r/(b+w). Numerator r > w.
Denominator (b+w) is less than (b+r).
if denominator is constant, greater the numerator greater the fraction
if numerator is constant, lower the numerator greater the fraction
In this case, the numerator is greater, combined with denominator being lesser makes the fraction larger.
Example: consider b=3
2>1, so(3+2)>(3+1)
Hence, 2/(3+1) > 1/(3+2).

Cheers!

Things are not what they appear to be... nor are they otherwise

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