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There are 10 books on a shelf: 5 English books, 3 Spanish bo

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by BTGmoderatorDC » Mon Dec 24, 2018 7:25 pm

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There are 10 books on a shelf: 5 English books, 3 Spanish books and 2 Portuguese books. What is the probability of choosing 2 books in different languages?

A. 31/90
B. 3/10
C. 1/3
D. 31/45
E. 28/90

OA D

Source: Economist Gmat
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by fskilnik@GMATH » Tue Dec 25, 2018 1:52 am
BTGmoderatorDC wrote:There are 10 books on a shelf: 5 English books, 3 Spanish books and 2 Portuguese books. What is the probability of choosing 2 books in different languages?

A. 31/90
B. 3/10
C. 1/3
D. 31/45
E. 28/90
Source: Economist Gmat
$$10\,\,{\rm{books}}\,\,{\rm{in}}\,\,3\,\,{\rm{langs}}\,\,\,\left\{ \matrix{
5\,\,{\rm{engl}} \hfill \cr
{\rm{3}}\,\,{\rm{span}} \hfill \cr
2\,\,{\rm{port}} \hfill \cr} \right.$$
$$? = P\left( {{\rm{2}}\,{\rm{langs}}\,\,{\rm{in}}\,\,{\rm{2}}\,\,{\rm{extractions}}} \right)$$
$${\rm{total}} = C\left( {10,2} \right) = {{10 \cdot 9} \over 2} = 45\,\,\,{\rm{equiprobables}}$$
$${\rm{favorable}}\,\, = \,\,\underbrace {5 \cdot 3}_{{\rm{engl}}\,\,\& \,\,{\rm{span}}} + \underbrace {5 \cdot 2}_{{\rm{engl}}\,\,\& \,\,{\rm{port}}} + \underbrace {3 \cdot 2}_{{\rm{span}}\,\,\& \,\,{\rm{port}}} = 31$$
$$? = {{31} \over {45}}$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
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by GMATGuruNY » Tue Dec 25, 2018 5:06 am
BTGmoderatorDC wrote:There are 10 books on a shelf: 5 English books, 3 Spanish books and 2 Portuguese books. What is the probability of choosing 2 books in different languages?

A. 31/90
B. 3/10
C. 1/3
D. 31/45
E. 28/90
P(2 different languages) = 1 - P(2 of the same language).

Case 1: P(2 English books are selected)
P(1st book is English) = 5/10. (Of the 10 books, 5 are English.)
P(2nd book is English) = 4/9. (Of the 9 remaining books, 4 are English.)
To combine these probabilities, we multiply:
5/10 * 4/9 = 20/90.

Case 2: P(2 Spanish books are selected)
P(1st book is Spanish) = 3/10. (Of the 10 books, 3 are Spanish.)
P(2nd book is Spanish) = 2/9. (Of the 9 remaining books, 2 are Spanish.)
To combine these probabilities, we multiply:
3/10 * 2/9 = 6/90.

Case 3: P(2 Portuguese books are selected)
P(1st book is Portuguese) = 2/10. (Of the 10 books, 2 are Portuguese.)
P(2nd book is Portuguese) = 1/9. (Of the 9 remaining books, 1 is Portuguese.)
To combine these probabilities, we multiply:
2/10 * 1/9 = 2/90.

P(2 of the same language) = Case 1 + Case 2 + Case 3 = 20/90 + 6/90 + 2/90 = 28/90 = 14/45.
P(2 different languages) = 1 - 14/45 = 31/45.

The correct answer is D.
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by Scott@TargetTestPrep » Fri Feb 15, 2019 5:17 pm
BTGmoderatorDC wrote:There are 10 books on a shelf: 5 English books, 3 Spanish books and 2 Portuguese books. What is the probability of choosing 2 books in different languages?

A. 31/90
B. 3/10
C. 1/3
D. 31/45
E. 28/90

OA D

Source: Economist Gmat
We can use the formula:

P(choosing two books of different language) = 1 - P(choosing two books of same language)

So our options are 2 English, 2 Spanish or 2 Portuguese books.

2 English books can be chosen in:

5C2 = (5 x 4)/2! = 10 ways

2 Spanish books can be chose in:

3C2 = 3 ways

2 Portuguese books can be chosen in:

2C2 = 1 way

The total ways to select 2 books from 10 is:

10C2 = (10 x 9)/2 = 45 ways

So P(choosing two books of same language) = (3 + 10 + 1)/45 = 14/45.

Thus, P(choosing two books of different language) = 1 - 14/45 = 31/45.

Alternate Solution:

If the first book is an English book (for which there is a 5/10 = 1/2 probability), the second book can be any of the 3 + 2 = 5 books among the 9 remaining books. Therefore, the probability of choosing an English book followed by a book in a different language is 1/2 x 5/9 = 5/18.

If the first book is a Spanish book (for which there is a 3/10 probability), the second book can be any of the 5 + 2 = 7 books among the 9 remaining books. Therefore, the probability of choosing an Spanish book followed by a book in a different language is 3/10 x 7/9 = 21/90 = 7/30.

If the first book is a Portuguese book (for which there is a 2/10 = 1/5 probability), the second book can be any of the 5 + 3 = 8 books among the 9 remaining books. Therefore, the probability of choosing an Portuguese book followed by a book in a different language is 1/5 x 8/9 = 8/45.

Therefore, the total probability of choosing two books in different languages is 5/18 + 7/30 + 8/45 = 25/90 + 21/90 + 16/90 = 62/90 = 31/45.

Answer: D

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