A bag contains a total of 20 only red and white marbles, fewer than

Question

A bag contains a total of 20 only red and white marbles, fewer than half of which are red. Two marbles are to be drawn simultaneously from the bag. How many marbles in bag are red?

(1) The probability that the two marbles to be drawn will be red is 1/19.
(2) The probability that one marble to be drawn will be red and the other will be white is 15/19.

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hr1212 · Answer

r + w = 20
r < 10
r?

Statement 1 - rC2/20C2 = 1/19
(r*(r-1))/(20*19) = 1/19
r*(r-1) = 20
r = 5 or r = -4

As r needs to be positive, r = 5. Hence sufficient.

Statement 2 - rC1*wC1/20C2 = 15/19
(r*w)/(20*19/2) = 15/19
r*w = 15*10
r*(20-r) = 150
r^2 - 20r + 150 = 0

As b^2 - 4ac < 0, no integer solutions exist for this quadratic. Hence, insufficient.

IMO: A

Bunuel , Unless I missed something, is the probability value correct for Statement 2 as per GMAT standard? I haven't seen DS questions failing for non-real roots as mostly they fail for out of scope values.

sujoykrdatta · Answer

r (red) + w (white) = 20; r < 10
We need to find the vlaue of r

S1: p(both red) = rC2/20C2 = r(r-1)/380 
=> r(r-1)/380 = 1/19 => r(r-1) = 20 => r = 5 (only viable solution) -  Sufficient

S2: p(one red and one white) = rC1 * (20-r)C1 / 20C2 = r(20-r)/190 
=> r(20-r)/190 = 15/19 => r(20-r) = 150. 
However, for r(20-r), the max value would be when r = 10, making ech term 10, hence having product 100. Thus, there cannot be a solution to this equation, implying that the statement itself is wrong
## I feel that in a proper DS question, we need to assume that the information in the statements is correct and on that basis, check if a solution is possible. Here, however, the statement itself is wrong. 
If we change the statement to "  The probability that one marble to be drawn will be red and the other will be white is less than 15/19  ", then it will be fine since that will be true for all possible values of r and the statement will be insufficient.

Shin0099 · Answer

(1) - Since the percentage for two marbles in a row is given, the amount can be calculated 20/380 = 5/20 * 4/19 -> 5 marbles, sufficient. 
(2) - Only option here is, that we have the same amount of marbles - Otherwise, we cant match number and colour. 
Here the leanest way should be to look for a possible sqare root in the nominator. since 300 (300/380) doesn't offer one, we can be sure, there arent 10 marbles of the same color.

For the same number of marbles counts (for each of the two possible grabs): 10/20*10/19 = 100/380 = 5/19. sqrt(100) = 10. Probability for both: (200/380)
For the given amount: 300/380/2 = 150/380 n/20 * n-1/19 = not an integer.

So two is not sufficient.
I may did a mistake here. Please correct me!. While the argumentation for the insufficiency should hold, the proof didn't.

That results IMO in A.

NoeticImbecile · Answer

Hello, How is S2 correct?

A bag contains a total of 20 only red and white marbles, fewer than

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