Problem Solving

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]

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

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

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.