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# Lily has only red and blue balls in a jar. If she removes three balls,

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SC Moderator
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Lily has only red and blue balls in a jar. If she removes three balls,  [#permalink]

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10 Jun 2018, 07:21
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5
00:00

Difficulty:

95% (hard)

Question Stats:

32% (02:11) correct 68% (02:40) wrong based on 98 sessions

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Lily has only red and blue balls in a jar. If she removes three balls, is the probability of getting all red balls greater than the probability of getting at least one blue ball?

(1) The number of red balls is more than three times the number of blue balls.
(2) Less than (1)/(4) th of the balls in the jar are blue.

Source: Expert Global

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Thanks!
Do give some kudos.

Simple strategy:
“Once you’ve eliminated the impossible, whatever remains, however improbable, must be the truth.”

GMAT Ninja YouTube! Series 1| GMAT Ninja YouTube! Series 2 | How to Improve GMAT Quant from Q49 to a Perfect Q51 | Time management

My Notes:
Reading comprehension | Critical Reasoning | Absolute Phrases | Subjunctive Mood
GMATH Teacher
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Lily has only red and blue balls in a jar. If she removes three balls,  [#permalink]

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29 Aug 2018, 06:57
1
Quote:
Lily has only red and blue balls in a jar. If she gets three balls randomly, is the probability of getting all red balls greater than the probability of getting at least one blue ball?

(1) The number of red balls is more than three times the number of blue balls.
(2) Less than (1)/(4) of the balls in the jar are blue.

Source: Expert Global

Nice problem!

$$\operatorname{P} \left( {{\text{all}}\,\,3\,\,{\text{red}}} \right)\,\,\,\,\mathop > \limits^? \,\,\,\operatorname{P} \left( {{\text{not}}\,\,{\text{all}}\,\,3\,\,{\text{red}}} \right)\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\operatorname{P} \left( {{\text{all}}\,\,3\,\,{\text{red}}} \right) = \boxed{\frac{{C\left( {r,3} \right)}}{{C\left( {r + b,3} \right)}}\,\,\,\,\mathop > \limits^? \,\,\,\frac{1}{2}}$$

$$\left( {1 + 2} \right)\,\,$$

$$Take\,\,\left( {r,b} \right) = \left( {4,1} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\frac{{C\left( {4,3} \right)}}{{C\left( {5,3} \right)}} = \,\frac{1}{{10}}\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\langle {{\text{NO}}} \right\rangle$$

$$Take\,\,\left( {r,b} \right) = \left( {5,1} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\frac{{C\left( {5,3} \right)}}{{C\left( {6,3} \right)}} = \,\frac{{10}}{{20}}\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\langle {{\text{NO}}} \right\rangle \,\,\,\,\,\,\,\,{\text{but}}\,\,\,{\text{increasing}}\,\,...$$

$$Take\,\,\left( {r,b} \right) = \left( {6,1} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\frac{{C\left( {6,3} \right)}}{{C\left( {7,3} \right)}} = \,\frac{4}{7}\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle \,\,\,\,\,\,\,\,$$

(Keeping the number of blue balls fixed and increasing the number of red ones, we know the probability of getting 3 reds must increase...)

The answer is (E), because we were able to BIFURCATE (1+2)!

The above follows the notations and rationale taught in the GMATH method.

Regards,
fskilnik.
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Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
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Re: Lily has only red and blue balls in a jar. If she removes three balls,  [#permalink]

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10 Jun 2018, 08:50
Answer should be E. Eg: Take there are 4 red balls and 1 blue ball. Then P(RRR)= 2/5 and P(atleast 1B) = 3/5. So 2nd expression is smaller.

Then take there are 7 red balls and 1 blue ball. Then P(RRR)= 5/8 and P(atleast 1B) = 3/8. So 1st expression is smaller.

This example will work for both the statements. Therefore, no sufficiency in any. Hence, E.
Intern
Joined: 03 Sep 2015
Posts: 18
Re: Lily has only red and blue balls in a jar. If she removes three balls,  [#permalink]

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28 Aug 2018, 10:13
Hi. How did you get 3/5 for blue balls?
Intern
Joined: 09 Aug 2018
Posts: 3
Re: Lily has only red and blue balls in a jar. If she removes three balls,  [#permalink]

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29 Aug 2018, 05:37
1
Hi
I find the resource helpful, you can also try this probability lecture.
Intern
Joined: 03 Sep 2015
Posts: 18
Re: Lily has only red and blue balls in a jar. If she removes three balls,  [#permalink]

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29 Aug 2018, 05:48
Thanks so much for this

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Re: Lily has only red and blue balls in a jar. If she removes three balls,   [#permalink] 29 Aug 2018, 05:48
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