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# A box contains 10 light bulbs, fewer than half of which are defective.

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A box contains 10 light bulbs, fewer than half of which are defective. [#permalink]
CRACKGMATNUT wrote:
Bunuel wrote:
udaymathapati wrote:
A box contains 10 light bulbs, fewer than half of which are defective. Two bulbs are to be drawn simultaneously from the box. If n of the bulbs in box are defective, what is the value of n?
(1) The probability that the two bulbs to be drawn will be defective is 1/15.
(2) The probability that one of the bulbs to be drawn will be defective and the other will not be defective is 7/15.

Given: $$bulbs=10$$ and $$defective=n<5$$. Question: $$n=?$$

(1) The probability that the two bulbs to be drawn will be defective is 1/15 --> clearly sufficient, as probability, $$p$$, of drawing 2 defective bulbs out of total 10 bulbs, obviously depends on # of defective bulbs, $$n$$, so we can calculate uniques value of $$n$$ if we are given $$p$$.

To show how it can be done: $$\frac{n}{10}*\frac{n-1}{9}=\frac{1}{15}$$ --> $$n(n-1)=6$$ --> $$n=3$$ or $$n=-2$$ (not a valid solution as $$n$$ represents # of defective bulbs and can not be negative). Sufficient.

(2) The probability that one of the bulbs to be drawn will be defective and the other will not be defective is 7/15 --> also sufficient, but a little bit trickier: if it were 3 defective and 7 good bulbs OR 7 defective and 3 good bulbs, then the probability of drawing one defective and one good bulb would be the same for both cases (symmetric distribution), so info about the probability, 7/15, of drawing one defective and one good bulb would give us 2 values of $$n$$ one less than 5 and another more than 5 (their sum would be 10), but as we are given that $$n<5$$, we can stiil get unique value of $$n$$ which is less than 5.

To show how it can be done: $$2*\frac{n}{10}*\frac{10-n}{9}=\frac{7}{15}$$ --> $$n(10-n)=21$$ --> $$n=3$$ or $$n=7$$ (not a valid solution as $$n<5$$). Sufficient.

Hope it's clear.

Hi Beunel,

Can you please tell me in evaluating option B, Why did you multiply the equation by 2?

I tried explaining this in this post on page 1. Let me know if still unclear.
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A box contains 10 light bulbs, fewer than half of which are defective. [#permalink]
Bunuel wrote:
Hi Beunel,

Can you please tell me in evaluating option B, Why did you multiply the equation by 2?

I tried explaining this [ur=https://gmatclub.com/forum/a-box-contains-10-light-bulbs-fewer-than-half-of-which-are-defective-99940.html#p1072120l]in this post on page 1[/url]. Let me know if still unclear.

Hi Bunuel,

Can I ask, that we can always multiply by 2 in situations like these, irrespective of the value of n? in other words, would the outcomes in such cases always be symmetric, or are there situations where they may not be symmetric?
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Re: A box contains 10 light bulbs, fewer than half of which are defective. [#permalink]
Hi.
Maybe my method helps !
I didnt let the equation turn into a quadratic one.
Once I had massaged the question well, i knew that n<=4(defective bulbs) and the non-defective bulbs were (10-n).

Moving to statements now:-
Statement 1- Boiled down to (n)(n-1)=6.
I already know that the values of n could be 1,2, 3 and 4.
But only one value of n satisfies this equation i.e. 3.

Statement 2 – boiled down to (n)(10-n)=21
Again, only for n=3, the statement could hold true.

Side note- In DS questions Statements will never contradict each other. Once I have figured, from statement 1, that the value of n is 3, I can use this as a "hint" while i am trying to figure out value of n in statement 2.
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Re: A box contains 10 light bulbs, fewer than half of which are defective. [#permalink]