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A certain jar contains only b black marbles, w white marbles  [#permalink]

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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 then 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: a certain jar contains  [#permalink]

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anilnandyala 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 then the probability that the marble chosen will be white?

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

The question is $$\frac{R}{R+B+W}>\frac{W}{R+B+W}$$ true? Or is $$R>W$$ true?

(1) $$\frac{R}{B+W} > \frac{W}{B+R}$$ --> $$\frac{R}{B+W} +1> \frac{W}{B+R}+1$$ --> $$\frac{R+B+W}{B+W}> \frac{W+B+R}{B+R}$$ --> $$\frac{1}{B+W}> \frac{1}{B+R}$$ --> $$B+R>B+W$$ --> $$R>W$$. Sufficient.

OR:
Given: $$\frac{R}{B+W} > \frac{W}{B+R}$$ -->

Cross multiply, we can safely do this as $$B+W$$ and $$B+R$$ are more than zero.

We'll get $$R(B+R)>W(B+W)$$ --> $$RB+R^2>WB+W^2$$ --> $$(R^2-W^2)+(RB-WB)>0$$ --> $$(R-W)(R+W)+B(R-W)>0$$ --> $$(R-W)(R+W+B)>0$$.

As $$R+W+B>0$$, the above inequality to hold true $$R-W$$ must also be more than zero, so $$R-W>0$$ --> $$R>W$$.

(2) $$B-W>R$$, not sufficient to determine whether $$R>W$$ or not.

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Re: a certain jar contains  [#permalink]

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anilnandyala wrote:
a certain jar contains only b black marbles, w white marbles & r red marbles. if one marble is to be chosen random from jar is the probability that the marble chosen will be red greater then the probability the marble chosen is white?
a) r/(b+w) > w/(b+r)
b) b-w > r

The probability that red marble is chosen will be greater than the probability that white marble is chosen if there are more red marbles than white marbles.
So the queestion is just: Is r > w

Statement 1: r/(b + w) > w/(b + r)
Cross multiply to get r(b + r) > w(b + w) .... [(b + w) and (b + r) are definitely positive so cross multiplying is not a problem.]
Now, if r > w, (b + r) has to be greater than (b + w)
If r were less than w, then (b + r) < (b + w) and the left side would have been smaller than the right side.
So this implies that r must be greater than w. Sufficient.

Statement 2: b > r + w
But we cant compare r and w so not sufficient.

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Re: a certain jar contains  [#permalink]

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VeritasPrepKarishma and Bunuel - thanks a lot for ur explanations.

+1 from me... again.

keep up the good job.
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Re: a certain jar contains  [#permalink]

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Bunuel wrote:
anilnandyala wrote:
a certain jar contains only b black marbles, w white marbles & r red marbles. if one marble is to be chosen random from jar is the probability that the marble chosen will be red greater then the probability the marble chosen is white?
a) r/(b+w) > w/(b+r)
b) b-w > r

(1) $$\frac{R}{B+W} > \frac{W}{B+R}$$ --> $$\frac{R}{B+W} +1> \frac{W}{B+R}+1$$ --> $$\frac{R+B+W}{B+W}> \frac{W+B+R}{B+R}$$ --> $$\frac{1}{B+W}> \frac{1}{B+R}$$ --> $$B+R>B+W$$ --> $$R>W$$. Sufficient.

OR:
Given: $$\frac{R}{B+W} > \frac{W}{B+R}$$ -->

Cross multiply, we can safely do this as $$B+W$$ and $$B+R$$ are more then zero.

awesome explanation. I have translated the question to is r>w but did not know how to solve the (1). Now get it clear
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A certain jar contains only b black marbles, w white marbles and  [#permalink]

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lets rephrase the question first.
It says IS r/(w+r+b)> w(r+w+b)
Cross multiply because we know all the variables are positive. It becomes
Is br+r^2> bw+w^2 ?

Statement 1: when cross multiplying we ger br+ r^2 >bw+ w^2........Thus sufficient

Statement 2: clearly insufficient.

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Re: A certain jar contains only b black marbles, w white marbles  [#permalink]

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3
chose B. I hate my life!
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A jar contains 8 red marbles and y white marbles. If Joan takes  [#permalink]

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Given 8 red marbles and y white marbles.
Number of ways you can pick any two marbles is $$(8+Y)C2$$.
Ways of picking 2 red marbles is 8*7 (first time you can pick out of 8 red marbles and second time you can pick one of the remaining 7 marbles).
Ways of picking 1 marble of each color. This can happen in 2 ways. 1 way) First pick white and second pick red [y*8 ways] or 2 way) first pick red and second pick white [8*y ways]. So total number of ways to pick two different colors will be $$y*8 + 8*y$$ = $$2*8y$$

A jar contains 8 red marbles and y white marbles. If Joan takes 2 random marbles from the jar, is it more likely that she will have 2 red marbles than that she will have one marble of each color?

Question is asking for - is probability of picking 2 red marbles > probability of picking different color marbles, is $$\frac{8*7}{(8+Y)C2} > \frac{2*8y}{(8+Y)C2}$$
Therefore, $$y<7/2=3.5$$

(1) y ≤ 8 y can be less than or greater than 3.5 - NS
(2) y ≥ 4 y is always greater than 3.5 - S

So B
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Re: A certain jar contains only b black marbles, w white marbles  [#permalink]

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1
anilnandyala 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 then the probability that the marble chosen will be white?

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

Is r>w?

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

rb + r^2 ? wb + w^2

(r+w)(r-w) > b(w-r)

Now if r-w>0 then w-r < 0 and inequality holds true.
Other way around if r-w<0, LHS is negative and RHS is positive and inequality does NOT hold true

So only valid scenario is r-w>0 and thus r>w

Sufficient

(B) Not enough Info

Cheers
J PS. Alternatively, first statement can be treated as

r-w / b+r>0

Well b+r is always positive, thus r-w has to be positive too
Then r>w
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Re: A certain jar contains only b black marbles, w white marbles  [#permalink]

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let's rephrase the question as "Is no of red marbles > no of white marbles ?"

Stmt1: r/(b+w) > w/(b+r)

There are two possibilities arising from the above statement .

A fraction r/(b+w) is greater than other fraction w/(b+r), if and only if the following 2 conditions are satisfied.
a) r>w or b) (b+r) >( b+w)
case1 : numerator 'r' is larger than numerator 'w';
case2 : denominator (b+r) > denominator (b+w) ;
in either case we get r>w ;
hence sufficient;

Stmt2: b-w > r

We can consider two cases(numbers) for which the above statement is both true and false;

case1: b=10; w=2 ; r=7;
10-2>7;
r>w(true)

Case2: b=10; w=5; r=4;
10-5>4;
but here 4>5(false);
Hence Insufficient.
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A certain jar contains only b black marbles, w white marbles  [#permalink]

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9
4
I like this problem because there are at least five different ways to solve it. I'll mention a more conceptual solution since no one has mentioned it yet, but there are some great solutions above as well:

If you know the concept of "odds" that is used in daily life, you can answer this question very quickly. "Odds" are just ratios of good outcomes to bad outcomes, while probabilities are ratios of good outcomes to total outcomes (good+bad). So when we say the odds that something will happen are 2 to 1, that means there's a 2/3 probability it will happen, and a 1/3 probability it will not.

In this question, the fraction r/(b+w) is just the ratio of red marbles to other marbles, so it just represents the odds of picking a red marble. Similarly the fraction w/(b+r) is the ratio of white marbles to other marbles, so it represents the odds of picking a white marble. And if the odds of getting red are better than the odds of getting white, the probability of getting red must be higher than the probability of getting white, so S1 is sufficient.
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A certain jar contains only b black marbles, w white marbles  [#permalink]

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IanStewart wrote:
I like this problem because there are at least five different ways to solve it. I'll mention a more conceptual solution since no one has mentioned it yet, but there are some great solutions above as well:

If you know the concept of "odds" that is used in daily life, you can answer this question very quickly. "Odds" are just ratios of good outcomes to bad outcomes, while probabilities are ratios of good outcomes to total outcomes (good+bad). So when we say the odds that something will happen are 2 to 1, that means there's a 2/3 probability it will happen, and a 1/3 probability it will not.

In this question, the fraction r/(b+w) is just the ratio of red marbles to other marbles, so it just represents the odds of picking a red marble. Similarly the fraction w/(b+r) is the ratio of white marbles to other marbles, so it represents the odds of picking a white marble. And if the odds of getting red are better than the odds of getting white, the probability of getting red must be higher than the probability of getting white, so S1 is sufficient.

This is exactly how I solved it. I just didnt realize I was using the concept of odds And glad to see you are still active here on the gmatclub IanStewart! Thanks
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Re: A certain jar contains only b black marbles, w white marbles  [#permalink]

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Bunuel wrote:
anilnandyala 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 then the probability that the marble chosen will be white?

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

The question is $$\frac{R}{R+B+W}>\frac{W}{R+B+W}$$ true? Or is $$R>W$$ true?

(1) $$\frac{R}{B+W} > \frac{W}{B+R}$$ --> $$\frac{R}{B+W} +1> \frac{W}{B+R}+1$$ --> $$\frac{R+B+W}{B+W}> \frac{W+B+R}{B+R}$$ --> $$\frac{1}{B+W}> \frac{1}{B+R}$$ --> $$B+R>B+W$$ --> $$R>W$$. Sufficient.

I know this is a very old post. Apologies. But why did you add 1 on both sides?
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Re: A certain jar contains only b black marbles, w white marbles  [#permalink]

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shahidhussaink wrote:
Bunuel wrote:
anilnandyala 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 then the probability that the marble chosen will be white?

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

The question is $$\frac{R}{R+B+W}>\frac{W}{R+B+W}$$ true? Or is $$R>W$$ true?

(1) $$\frac{R}{B+W} > \frac{W}{B+R}$$ --> $$\frac{R}{B+W} +1> \frac{W}{B+R}+1$$ --> $$\frac{R+B+W}{B+W}> \frac{W+B+R}{B+R}$$ --> $$\frac{1}{B+W}> \frac{1}{B+R}$$ --> $$B+R>B+W$$ --> $$R>W$$. Sufficient.

I know this is a very old post. Apologies. But why did you add 1 on both sides?

Let us not add 1 on both sides and do something totally berserk. Bear with me GMAT has driven me half nuts ok we need to find if R>W
Statement 2 is invalid it gives us nothing about R and W it just says B is more than R and W taken together. Forget that distraction.

Now coming hereto statement1
it says

R/W+b>W/B+R
Now according to the rules of ratio we can add or substract a constant from both numerator and denominator
Hence

R+W+B/2(W+B)> R+W+B/2(B+R)

We simply added the denominator to both the numerator and denominator. With me till here?

Ok

Now simplify

1/w+b>1/B+R

fine ? now we can cross multiple. They are balls so cannot be negative.
B+R> B+W
B cancels out
R>W Proved!
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Re: A certain jar contains only b black marbles, w white marbles  [#permalink]

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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 then the probability that the marble chosen will be white?

(1) r/(b+w) > w/(b+r)
add wr in both side and it leads to solution. Yes.
Suff.

(2) b-w > r
b > w + r

No info on w and r .

NS.
Ans A.
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Re: A certain jar contains only b black marbles, w white marbles  [#permalink]

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Top Contributor
anilnandyala 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 then the probability that the marble chosen will be white?

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

Target question: Is the probability that the marble chosen will be red greater than the probability that the marble chosen will be white?

We can rephrase the target question as...
REPHRASED target question: Is r > w?

Statement 1: r/(b + w) > w/(b + r)
Let's let T = the TOTAL number of marbles in the jar.
This means that b + w + r = T
This also means that b + w = T - r
And it means that b + r = T - w
So, we can take statement 1, r/(b + w) > w/(b + r), and rewrite it as...
r/(T - r) > w/(T - w)
Multiply both sides by (T - r) to get: r > w(T - r)/(T - w)
Multiply both sides by (T - w) to get: r(T - w) > w(T - r)
Expand both sides: rT - rw > wT - rw
Add rw to both sides: rT > wT
Divide both sides by T to get: r > w
Since we can answer the REPHRASED target question with certainty, statement 1 is SUFFICIENT

Statement 2: b - w > r
Add w to both sides to get: b > w + r
All this means is that there are more black marbles than there are white and red marbles combined.
Given this information, there's no way to determine whether or not r is greater than w
Since we cannot answer the REPHRASED target question with certainty, statement 2 is NOT SUFFICIENT

Cheers,
Brent
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Re: A certain jar contains only b black marbles, w white marbles  [#permalink]

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Can someone help me where I am going wrong?

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

Cross multiplying:

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

rb + r^2 > wb + w^2

rb - wb > w^2 - r^2

b(r-w) > (w-r)(w+r)

-b(w-r) > (w-r)(w+r)

(w-r) gets cancelled on both sides.

-b > w + r

-(w+b) > r

r < -(w+b)

Now what do I do?
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Re: A certain jar contains only b black marbles, w white marbles  [#permalink]

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anilnandyala 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 then the probability that the marble chosen will be white?

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

Hi chetan2u

Is it really necessary to do the long simplifications for st 1 ?
$$\frac{r}{(b+w)} > \frac{w}{(b+r)}$$

Since we know "b" is just some constant, we can just start by taking the above as -
$$\frac{r}{(w)} > \frac{w}{(r)}$$

After which it simply is -
$$r^2 > w^2$$
or
$$r > w$$

Is this approach okay?
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Re: A certain jar contains only b black marbles, w white marbles  [#permalink]

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blitzkriegxX wrote:
anilnandyala 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 then the probability that the marble chosen will be white?

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

Hi chetan2u

Is it really necessary to do the long simplifications for st 1 ?
$$\frac{r}{(b+w)} > \frac{w}{(b+r)}$$

Since we know "b" is just some constant, we can just start by taking the above as -
$$\frac{r}{(w)} > \frac{w}{(r)}$$

After which it simply is -
$$r^2 > w^2$$
or
$$r > w$$

Is this approach okay?

Yes, this is much simpler and better..
You can do this as all variables are positive and the question finally boils down to which is more r or w..
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Re: A certain jar contains only b black marbles, w white marbles  [#permalink]

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Bunuel wrote:
anilnandyala 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 then the probability that the marble chosen will be white?

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

The question is $$\frac{R}{R+B+W}>\frac{W}{R+B+W}$$ true? Or is $$R>W$$ true?

(1) $$\frac{R}{B+W} > \frac{W}{B+R}$$ --> $$\frac{R}{B+W} +1> \frac{W}{B+R}+1$$ --> $$\frac{R+B+W}{B+W}> \frac{W+B+R}{B+R}$$ --> $$\frac{1}{B+W}> \frac{1}{B+R}$$ --> $$B+R>B+W$$ --> $$R>W$$. Sufficient.

OR:
Given: $$\frac{R}{B+W} > \frac{W}{B+R}$$ -->

Cross multiply, we can safely do this as $$B+W$$ and $$B+R$$ are more than zero.

We'll get $$R(B+R)>W(B+W)$$ --> $$RB+R^2>WB+W^2$$ --> $$(R^2-W^2)+(RB-WB)>0$$ --> $$(R-W)(R+W)+B(R-W)>0$$ --> $$(R-W)(R+W+B)>0$$.

As $$R+W+B>0$$, the above inequality to hold true $$R-W$$ must also be more than zero, so $$R-W>0$$ --> $$R>W$$.
(2) $$B-W>R$$, not sufficient to determine whether $$R>W$$ or not.

I don't understand the highlighted part. Would you please explain it in simple words? Bunuel

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