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# A bag contains at least 10 blue marbles and at least 10 red marbles

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Math Expert
Joined: 02 Sep 2009
Posts: 59674
A bag contains at least 10 blue marbles and at least 10 red marbles  [#permalink]

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25 Nov 2019, 01:02
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21% (01:42) correct 79% (01:38) wrong based on 21 sessions

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A bag contains at least 10 blue marbles and at least 10 red marbles, and contains marbles of no other color. Is the number of blue marbles in the bag greater than the number of red marbles?

(1) If two marbles are chosen from the bag without replacement, the probability of picking two blue marbles is higher than the probability of picking one blue marble and one red marble.

(2) If two marbles are chosen from the bag without replacement, the probability of picking one blue marble and one red marble is higher than the probability of picking two red marbles.

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Math Expert
Joined: 02 Aug 2009
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Re: A bag contains at least 10 blue marbles and at least 10 red marbles  [#permalink]

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26 Nov 2019, 19:49
A bag contains at least 10 blue marbles and at least 10 red marbles, and contains marbles of no other color. Is the number of blue marbles in the bag greater than the number of red marbles?

Let there be B blue marbles and R red marbles...
We have to find whether B>R

(1) If two marbles are chosen from the bag without replacement, the probability of picking two blue marbles is higher than the probability of picking one blue marble and one red marble.
The probability of picking two blue marbles = $$\frac{BC2}{(B+R)C2}$$
The probability of picking one blue marble and one red marble = $$\frac{BC1*RC1}{(B+R)C2}$$
So, $$BC2>BC1*RC1......B(B-1)>2BR......(B-1)>2R$$, so surely $$B>R$$
Suff

(2) If two marbles are chosen from the bag without replacement, the probability of picking one blue marble and one red marble is higher than the probability of picking two red marbles.
The probability of picking two red marbles = $$\frac{RC2}{(B+R)C2}$$
The probability of picking one blue marble and one red marble = $$\frac{BC1*RC1}{(B+R)C2}$$
So, $$RC2<BC1*RC1......R(R-1)<2BR......(R-1)<2B$$,
If R = 20 and B 15...NO
If R=20 and B=30..Yes
Insufficient

A
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Re: A bag contains at least 10 blue marbles and at least 10 red marbles   [#permalink] 26 Nov 2019, 19:49
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# A bag contains at least 10 blue marbles and at least 10 red marbles

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