M28-45

Question

All marbles in a jar containing 10 marbles are either red or blue. If two marbles are drawn from the jar, at random and without replacement, is the probability that both marbles are red greater than  \frac{3}{5} ?

(1) The probability that both marbles selected will be blue is less than  \frac{1}{10} .
 
(2) At least 60% of the marbles in the jar are red.

Bunuel · Accepted Answer

Official Solution:

All marbles in a jar containing 10 marbles are either red or blue. If two marbles are drawn from the jar, at random and without replacement, is the probability that both marbles are red greater than  \frac{3}{5} ? 
 
The question asks whether  P(R \ and \ R)=\frac{R}{10}*\frac{R-1}{9} \gt \frac{3}{5} , where R is the number of red marbles in the jar.
 
Is  R(R-1) \gt 54 ?
 
By plugging in values for  R , we find that, for the above inequality to be true,  R  must be greater than 7. Hence, the question asks whether  R > 7 , so whether the number of red marbles is 8, 9, or 10..
 
(1) The probability that both marbles selected will be blue is less than  \frac{1}{10} .
 
This implies that  \frac{B}{10}*\frac{B-1}{9} \lt \frac{1}{10} , where B is the number of blue marbles in the jar. Simplifying this inequality, we get  B(B-1) < 9 , By plugging in values for  B , we find that  B  must be less than 4. Therefore, the number of red marbles in the jar can be 7, 8, 9, or 10:  R > 6 . Not sufficient.
 
(2) At least 60% of the marbles in the jar are red. This means that the number of red marbles is greater than or equal to 6:  R \geq 6 . Not sufficient.
 
(1)+(2) Combining the two statements, we know that the number of red marbles in the jar is between 6 and 10:  R \gt 6 . If  R  is 6 or 7, answer is NO but if  R  is 8, 9, or 10, the answer is YES. Not sufficient.

Answer: E

shrutikasat · Answer

Bunnel

How r(r-1)>54 
 is 7?

can u please explain?

Bunuel · Answer

By number plugging. R is an integer, if R = 7, then R(R - 1) = 42 < 54 but if R = 8, then R(R - 1) = 56 > 54, thus R must be greater than 7.

swagatamoitra · Answer

I think this is a  high-quality  question and I agree with explanation. // good question // Approach is worth noting and understanding for future use //

pbad · Answer

I think this is a  high-quality  question and I agree with explanation. Very good question. Very clear and simple explanation.

akiara · Answer

I think this is a  high-quality  question and I agree with explanation.

akhtolkhyn · Answer

I think this is a  high-quality  question and I agree with explanation.

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

BottomJee · Answer

I think this is a  high-quality  question and I agree with explanation.

Rashmi96 · Answer

I like the solution - it’s helpful. what is the source of these questions

Bunuel · Answer

The source of the questions included in GMAT Club tests is GMAT Club.

crimson_noise · Answer

I don’t quite agree with the solution. If there are 10 marbles and probability of picking 2 blue marbles from it is <1/10, then blue marbles are either 0 or 1, which implies picking two marbles is 0.72 or 1, both are greater than 60%

Bunuel · Answer

You are wrong.

First of all, if there is 1 blue marble and 9 red marbles, how is the probability of picking two red marbles 0.72? It's 9/10 * 8/9 = 4/5 = 0.8, not 0.72.

Statement (1) says the probability of drawing two blue marbles is less than 1/10. As shown in the solution, this means there can be 0, 1, 2, or 3 blue marbles. For example, with 3 blue marbles, the probability of both being blue is 3/10 * 2/9 = 1/15, which is less than 1/10.

This means the number of red marbles could be 7 (which gives a NO answer) or 8, 9, or 10 (which give YES). So we cannot conclude definitively.

You should review and study the official solution more carefully.

Tushar777 · Answer

This is a great question that’s helpful for learning and I like the solution - it’s helpful.

navs18 · Answer

I like the solution - it’s helpful.

sarasmart · Answer

I did not quite understand the solution. How do you find out the number 54? And how do you find out the number 4?

Bunuel · Answer

1. We have  \frac{R}{10}*\frac{R-1}{9} \gt \frac{3}{5}

Multiply by 90:  R(R-1)\gt 54 .

2. We have  B(B-1) < 9

By plugging in values for B , we find that  B  must be less than 4.

Arjunnayak14 · Answer

This is a great question that’s helpful for learning.

Milletur · Answer

I like the solution - it’s helpful.

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