In a box, 20 bulbs are kept and some of them are defective. 2 bulbs

Question

In a box, 20 bulbs are kept and some of them are defective. 2 bulbs are selected randomly. Is the probability that both selected bulbs are defective less than 0.4?

(1) More than 50% of the bulbs are defective.
(2) Less than 60% of the bulbs are defective.

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CrackverbalGMAT · Accepted Answer

Hello,

Greetings for the day!

This is a question of medium difficulty level. The statements have been framed in such a way that if you do not examine them carefully, you will end up falling for the trap answer which is Option C.

Let us now look at how to solve this question:
Of the 20 bulbs, let ‘x’ bulbs be defective and ‘y’ be non-defective. 
If 2 bulbs are selected at random, the total possible outcomes will be 20C2. 
Now, 20C2 = (20 x 19)/2 = 190.
Number of ways to select two bulbs which are defective = xC2, which also represents the favourable outcomes for the event defined in the question.

Therefore, Probability (Both bulbs are defective) = xC2 /190 The question asks us if this probability is less than 0.4. 
Now, 0.4 = 2/5.

Hence, we are trying to ascertain if (xC2 / 190) < (2/5) which simplifies to whether 
xC2 < 76.

xC2 can be less than 76 only if x is less than 13. Now, this is the data that we have to look for in the statements.

Statement I says that more than 50% of the bulbs were defective. This means that x can be 11 or higher. This does not conclusively tell us if x is less than 13 or not. Hence, statement I alone is insufficient.

Statement II says that less than 60% of the bulbs are defective. This means that x is 12 or lesser. If x is 12 or lesser, it is definitely lesser than 13, which is what we were trying to ascertain. Hence, statement II alone is sufficient.

Hence, the correct answer option is B.

In such DS questions on probability, it is important to analyse the question as much as possible and create a mathematical situation wherein you can use the data given in the statements with ease, as we did in this question. If you do not utilize the question data to your advantage, you will end up resorting to plugging in values which is not the ideal method for such questions.

Hope this helps!
Cheers.

ocelot22 · Answer

IF YOU FIND MY SOLUTION TO BE HELPFUL, PLEASE GIVE ME KUDOS (1) More than 50% of bulbs are defective. This means that more than 10 of the 20 bulbs are defective. Then the probability that two bulbs in a row will be defective is >10/20*9/20 = > 90/400 which is a little less than .25. So we can say the probability two bulbs are defective is > about .2 Of course there are several values >.2 that are less than and greater than .4 NS (2) Less than 60% of bulbs are defective. This means that < 12/20 are defective. Then the probability both bulbs are defective is <12/20 * 11/20 < 132/400, which is a little more than .25, lets say .3. Therefore the probability that both bulbs are defective < .3, which of course <.4 Sufficient The answer is B

EgmatQuantExpert · Answer

Solution 
  Given:  
In this question, we are given
 • In a box, 20 bulbs are kept and some of them are defective.
• 2 bulbs are randomly selected from the box.

To find:  
We need to determine
 • If the probability that both selected bulbs are defective is less than 0.4 or not.

Approach and Working:  
Let us assume that among 20 bulbs, N number of bulbs are defective.
 • Therefore, the probability that 2 selected balls will be defective =  ^NC_2/^{20}C_2 = \frac{N(N – 1)}{2} * \frac{2}{19 * 20} = \frac{N(N – 1)}{19 * 20}

Now, if the probability is to be less than 0.4, then we can say
 •  \frac{N(N – 1)}{19 * 20} < 0.4 
Or, N(N – 1) < 152

As N is an integer, N must be less than 13.
Hence, we need to know whether N < 13 or not.

Analysing Statement 1  
As per the information given in statement 1, more than 50% of the bulbs are defective.
 • Therefore, we can say N > 10
• However, we cannot conclude whether N < 13 or not.

Hence, statement 1 is not sufficient to answer the question.

Analysing Statement 2  
As per the information given in statement 2, less than 60% of the bulbs are defective.
 • Therefore, we can say that N < 12
• As N < 12, we can definitely say N < 13.

Hence, statement 2 is sufficient to answer the question.

Hence, the correct answer is option B.

Answer: B

PreiteeRanjan · Answer

Always approach such type qns from option point of view. QN says whether p(xc2)<0.4? option-1 says (>50% are defective),implying either 11c2/20c2,which is <0.3,however on the contrary if we take 19c2/20c2,which is more than >0.4(so insuficient to answer) option-2 says <60% defective,implying <12 are defective.now take probability,which is lets take max value less than 12 as 11,11c2/20c2,which is less than <0.4)so sufficient. TAKES LESSER TIME COMPARED TO PREVIOUS THOUGHTS.

mike2j · Answer

Where are the answer options for A,B,C,D?

Bunuel · Answer

Hi,

This is a data sufficiency question. Options for DS questions are always the same and usually omitted on the site.

The data sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements, plus your knowledge of mathematics and everyday facts (such as the number of days in July or the meaning of the word counterclockwise), you must indicate whether—

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
D. EACH statement ALONE is sufficient to answer the question asked.
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

I suggest you to go through the following post  ALL YOU NEED FOR QUANT .

Hope this helps.

rushimehta · Answer

Bunuel   EgmatQuantExpert

Can someone please help me out here... 
when we say "2 bulbs are selected randomly", do we mean with replacement, or without replacement ?

In the OG question - "https://gmatclub.com/forum/a-box-of-light-bulbs-contains-exactly-3-light-bulbs-that-are-defective-305939.html?srsltid=AfmBOopH6aT13c9K1GFCX8rM6y7OIqHGck0MXBcOl35FCnszLUyxSNj6" even, when it has been mentioned that "a sample of light bulbs picked at random from this box"... we need to know the the method of picking (with or without replacement).

Why have we assumed in this question that the the method of picking is without replacement...

Please let me know where I am going wrong.

Thank you in advance!

In a box, 20 bulbs are kept and some of them are defective. 2 bulbs

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GMAT Data Sufficiency (DS) Questions

In a box, 20 bulbs are kept and some of them are defective. 2 bulbs

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