Each of the 45 books on a shelf is written either in English or in Spa

The maximum number of books that can be written in Spanish ( irrespective of whether its hardcover or paperback) is 15 as per option 2
So, lets say all spanish books are ppaerbacks ,then whats the probability ?? 15/45 i.e 1/3 <1/2
Hence B

For option A , we can't specifically determine how many spanish books are there
Hence insufficent

This is a 2 by 2 question that frequently appears in the GMAT Math.

Looking at the attachment, the question, if we modify, is d/45<1/2?, d<45/2=22.5?. Looking at the condition 1), we get d=15<22.5. The answer is yes and the condition is sufficient. Hence, the correct answer is B.


- Once we modify the original condition and the question according to the variable approach method 1, we can solve approximately 30% of DS questions.

Is there any further explanation on this? Not convinced at all on the OA.
Please help

Hi

Did you go through this solution:
https://gmatclub.com/forum/each-of-the- ... l#p1621801

Basically, as per second statement, only 15 books are in Spanish (15 is obviously less than half of 45). Even if all these Spanish books are paperback, then the probability of selecting a paperback Spanish book will be 15/45 = 1/3 which is < 1/2. And if some Spanish books are hardcover also, then the probability of selecing a Spanish paperback will be even lesser than 1/3 (so definitely lesser than 1/2). So, as per second statement, probability of picking a paperback Spanish book will always be < 1/2. That's why the answer is B.

Just a small correction , for statement 1) we can’t take none case bcz each book is either written in English or Spanish and each book is either a paperback or hardcover !!

We can take the last numbers like 1,2 instead of taking 0

Posted from my mobile device

Given: Each of the 45 books on a shelf is written either in English or in Spanish, and each of the books is either a hardcover book or a paperback. A book is to be selected at random from the books on the shelf.

Target question: Is the probability less than 1/2 that the book selected will be a paperback written in Spanish?
This is a good candidate for rephrasing the target question

Let's use the Double Matrix Method to solve this question. This technique can be used for most questions featuring a population in which each member has two characteristics associated with it (aka overlapping sets questions).
Here, we have a population of books, and the two characteristics are:
- English or Spanish
- Hardcover or paperback

We can set up our diagram like this:


1/2 of 45 = 22.5
So in order for the probability to be less than 1/2, we need 22 or fewer books to be Spanish paperbacks.
In other words, we need 22 or fewer books in the yellow shaded region.
Let's use this information to REPHRASE the target question
REPHRASED target question: Is the number of books in the yellow shaded region 22 or less?



Statement 1: Of the books on the shelf, 30 are paperback
If 30 of the 45 books are paperback, then the remaining 15 books must be hardcover. We'll add this to our diagram


There are many different ways to complete this diagram so that we meet the given conditions.
For example, we could have the following situation:

In this case, the answer to the REPHRASED target question is YES, the number of books in the yellow shaded region IS 22 or less

Alternatively, we could also have this situation:

In this case, the answer to the REPHRASED target question is NO, the number of books in the yellow shaded region is NOT 22 or less

Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: Of the books on the shelf, 15 are written in Spanish.
When we add this information to our diagram, we get:


IMPORTANT: This tells us that the two boxes in the bottom row, must add to 15.
So it will be impossible for the value in the yellow shaded region to be greater than 22.
In other words, the answer to the REPHRASED target question is YES, the number of books in the yellow shaded region IS 22 or less
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer: B

This question type is VERY COMMON on the GMAT, so be sure to master the technique.

To learn more about the Double Matrix Method, watch this video:


Cheers,
Brent

I have a doubt - As per stmt 2, if we say all 45 are paperback, then probability is 1/3; However, if we say 30 are paperback, wouldn't probability be 1/2, or for that matter 15 are paperback, wouldn't it make probability as 1 ?? Why shouldn't the answer be 'C'

Hello Anurag,

It seems like you have misinterpreted the question. The question is not about the probability of picking a Spanish paperback OF all the paperbacks.
The question asks us if the probability of picking a “Paperback written in Spanish” is less than ½ when ANY book is selected at random.
There’s as big a difference between the two as chalk from cheese.

This is a very common problem that many GMAT test takers face. They end up solving a drastically different question in their head when they have to actually solve the question they see on the screen. Look at the table given below:



The question says “If a BOOK is selected at random”. This means your sample space is 45 (the cell highlighted in Green) and hence should be in the denominator.
Whereas, the question that YOU solved in your head is “If a paperback is selected at random”. This means YOUR sample space was the cell highlighted in red, which is of course wrong.
I hope that clarifies your doubt.

Look at how many Spanish books are there in total, as per Statement II (the cell highlighted in blue). If there are 15 books of Spanish in total, there would be either 15 paperbacks or lesser. (the cell highlighted in yellow)
\(\frac{15}{45}\) is definitely less than ½. We can answer the question with a definite NO which is an accepted answer on a YES-NO type of DS question.

This is another common mistake I have seen test takers making on a YES-No type of DS question – forgetting that a NO is as good as a YES.
Since statement II alone is sufficient to give us a NO, the correct answer option is B.

Here’s a link to one of our articles on Data Sufficiency for your reference. This will really help you to not make such errors on a DS question.
https://bit.ly/348ndXx

Hope that helps!

Hi Aravind,

Thanks a lot for detailed explanation. It cleared my doubt. Thanks again.

This question throws just enough at you to be dangerous. This question also subtly tests your knowledge of probability. High quality question!

So all probabilities have to add to 1. Also, prob of (spanish and some other event) must be <= prob of spanish (total)

(1) of the books of the shelf 30 are PB. We know nothing of the number of Spanish or english books. if there are 0 spanish books the probability (PB and spanish) = 0 <1/2 so yes. if there are 45 spanish books, P(PB and spanish) =P(paperback) = 30/45 >1/2 so No NS

(2) Now, probability (spanish and paperback) < prob (spanish). Prob (spanish) =15/45 = 1/3 , so P(spanish and paperback) <1/3 <1/2 so Yes sufficient OA is B

IF you find my solution to be helpful, PLEASE GIVE ME KUDOS

My method:
1) 30 paperback/45*1 case (1 spanish/45)=6/405 yes
30/45*2 case (44/45)=264/405 -> 2/3 no
1 NS
2) 15 spanish/45*1 case (1 paperback/45)=1/135 yes
15/45*2 case (44/45)=132/405 -> 1/3 yes
2 Suff
Answer B

Posted from my mobile device

Each of the 45 books on a shelf is written either in English or in Spanish, and each of the books is either a hardcover book or a paperback. If a book is to be selected at random from the books on the shelf, is the probability less than 1/2 that the book selected will be a paperback written in Spanish?


(1) Of the books on the shelf, 30 are paperback.

(2) Of the books on the shelf, 15 are written in Spanish.

Stem: probability (spanish AND Paperback) = p(Spanish) * p(paperback) < 1/2

S1: paperback = 30
probability (spanish AND Paperback) = p(Spanish) * p(30/45) = p(Spanish) * 0.67 --> NS

S2: spanishbooks = 15
probability (spanish AND Paperback) = p(15/45) * p(paperback) = 0.33 * p(paperback)
now we know p(paperback) is less than 1
probability (spanish AND Paperback) = 0.33 * p(paperback) < 0.33 ---> Sufficient

OFFICIAL GMAT EXPLANATION

Arithmetic Probability

(1) This indicates that 30 of the 45 books are paperbacks. Of the 30 paperbacks, 25 could be written in Spanish. In this case, the probability of randomly selecting a paperback book written in Spanish is 25/45 > 1/2. On the other hand, it is possible that only 5 of the paperback books are written in Spanish. In this case, the probability of randomly selecting a paperback book written in Spanish is 5/45 < 12; NOT sufficient.

(2) This indicates that 15 of the books are written in Spanish. Then, at most 15 of the 45 books on the shelf are paperbacks written in Spanish, and the probability of randomly selecting a paperback book written in Spanish is at most 15/45 < 1/2; SUFFICIENT.

Statement 2 alone is sufficient.

I like this question a lot because we are typically used to seeing that a lot more info is needed to fill out the cells.

Each of the 45 books on a shelf is written either in English or in Spanish, and each of the books is either a hardcover book or a paperback. If a book is to be selected at random from the books on the shelf, is the probability less than 1/2 that the book selected will be a paperback written in Spanish?

(1) Of the books on the shelf, 30 are paperback.
This could mean one of two things:
Paperback + Spanish = 30 ----> 30/45 > 1/2 YES
Paperback + Spanish = 0 ----> 0/45 < 1/2 NO
Insufficient

(2) Of the books on the shelf, 15 are written in Spanish.
Paperback + Spanish = 15 ----> 15/45 < 1/2 YES
Paperback + Spanish = 0 ----> 0/45 < 1/2 YES

i.e. Since the maximum # of books that are paperback and Spanish is 15, it will never be the case that more than 50% of all 45 books is written in Spanish + paperback.

Answer is B.

Answer: Option B

Video solution by GMATinsight



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Statement 2 is alone sufficient.
Hence, OA is (B).

Video solution from Quant Reasoning:
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