probability

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Ramit88: Senior | Next Rank: 100 Posts; Posts: 67; Joined: Fri Nov 26, 2010 8:25 am; Thanked: 3 times

by Ramit88 » Sun Jan 23, 2011 8:45 am

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

[spoiler]ANS:A[/spoiler]

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Source: Beat The GMAT — Problem Solving | Sun Jan 23, 2011 8:45 am

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Rahul@gurome: GMAT Instructor; Posts: 1179; Joined: Sun Apr 11, 2010 9:07 pmFollowed by: 88 members; Location: Milpitas, CA; Thanked: 447 times

by Rahul@gurome » Sun Jan 23, 2011 9:17 am

Ramit88 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

Probability that the marble chosen will be red P(r) = r/(b + r + w)
Probability that the marble chosen will be white P(w) = w/(b + r + w)

Statement 1: r/(b + w) > w/(b + r)
Implies, (b + w)/r < (b + r)/w
Adding 1 to each side, 1 + (b + w)/r < 1 + (b + r)/w
=> (r + b + w)/r < (w + b + r)/w
=> r/(b + r + w) > w/(b + r + w)
=> P(r) > P(w)

Sufficient

Statement 2: (b - w) > r
Doesn't imply anything relevant.

Not Sufficient

The correct answer is A.

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Night reader: Legendary Member; Posts: 1337; Joined: Sat Dec 27, 2008 6:29 pmFollowed by: 10 members; Thanked: 127 times

by Night reader » Sun Jan 23, 2011 10:23 am

Ramit88 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

[spoiler]ANS:A[/spoiler]

# OUTCOMES is unknown, P(R) > P(W) with one selection ?
st(1) w(b+w)>r(b+r) --> wb+w^2-br-r^2>0 --> (w-r)(w+r) + b(w-r)>0 --> (w-r)(w+r+b)>0
w-r<0 AND w+r+b<0 --> cancel this branch as the total is less than 1
OR
w-r>0 AND w+r+b>0 --> w>r the, hence the number of white marbles is greater than the number of red marbles P(w)>P(r) Sufficient
st(2) b-w>r we don't know the number of white and red marbles, hence probabilistic model fails here Not sufficient

IOM A

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aleph777: Master | Next Rank: 500 Posts; Posts: 131; Joined: Fri Jun 18, 2010 10:19 am; Location: New York, NY; Thanked: 10 times

by aleph777 » Mon Jan 24, 2011 7:33 am

Nice problem.

First thing I did was break down the question. What is it really asking? Well, in order for the probability of red to be greater than the probability of white, the jar simply needs to have more red than white marbles.

So the question is really just asking is r > w?

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

Since we know all variables are positive, we can cross multiply:

r^2 +rb > w^2 + wb

Since the equation is doing the same thing on the left to red as it is to white on the right (squaring and then adding the product of r or w and blue), the equation prove that, all things similar, red is greater than white.

SUFFICIENT

STATEMENT 2:
b-w > r

No matter how we spin this, we can't learn anything about the relationship between r and w.

INSUFFICIENT

Answer: A

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Mon Jan 24, 2011 7:48 am

Ramit88 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

[spoiler]ANS:A[/spoiler]

Very little math is needed. Just use common sense.

Rewritten, the question is asking: Is R>W?

Statement 1:
Tells us that the ratio of R to the other marbles is greater than the ratio of W to the other marbles. Thus, R>W. Sufficient.

Statement 2:
Tells us that B > R+W. No way to determine whether R>W. Insufficient.

The correct answer is A.

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