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maggie27
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Mathematics, physics, and chemistry books are stored on a li 23 Jul 2014, 22:06
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C
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75% (hard)

Question Stats:

59% (02:50) correct 41% (02:56) wrong based on 338 sessions

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Mathematics, physics, and chemistry books are stored on a library shelf that can accommodate 25 books. Currently, 20% of the shelf spots remain empty. There are twice as many mathematics books as physics books and the number of physics books is 4 greater than that of the chemistry books. Ricardo selects 1 book at random from the shelf, reads it in the library, and then returns it to the shelf. Then he again chooses 1 book at random from the shelf and checks it out in order to read at home. What is the probability Ricardo reads 1 book on mathematics and 1 on chemistry?

A) 3%
B) 6%
C) 12%
D) 20%
E) 24%
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C
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KarishmaB
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Mathematics, physics, and chemistry books are stored on a li 24 Jul 2014, 00:31
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maggie27
Mathematics, physics, and chemistry books are stored on a library shelf that can accommodate 25 books. Currently, 20% of the shelf spots remain empty. There are twice as many mathematics books as physics books and the number of physics books is 4 greater than that of the chemistry books. Ricardo selects 1 book at random from the shelf, reads it in the library, and then returns it to the shelf. Then he again chooses 1 book at random from the shelf and checks it out in order to read at home. What is the probability Ricardo reads 1 book on mathematics and 1 on chemistry?

A) 3%
B) 6%
C) 12%
D) 20%
E) 24%
Show more

Shelf can accomodate 25 books but 20% of it is empty so there are in all 20 books.
If no of Chem books = x, number of Physics book = x+4, number of Math books = 2(x+4)

x + x+4 + 2(x+4) = 4x + 12 = 20
x = 2

Chem books = 2, Phy books = 6, Math books = 12

Probability of picking a Math book = 12/20
Probability of picking a Chem book = 2/20

Total Probability = Pick Math first and then Chem + Pick Chem first and then Math
= (12/20) * (2/20) + (2/20) * (12/20) = 12/100 = 12%
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Mathematics, physics, and chemistry books are stored on a li 30 Dec 2017, 01:12
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maggie27
Mathematics, physics, and chemistry books are stored on a library shelf that can accommodate 25 books. Currently, 20% of the shelf spots remain empty. There are twice as many mathematics books as physics books and the number of physics books is 4 greater than that of the chemistry books. Ricardo selects 1 book at random from the shelf, reads it in the library, and then returns it to the shelf. Then he again chooses 1 book at random from the shelf and checks it out in order to read at home. What is the probability Ricardo reads 1 book on mathematics and 1 on chemistry?

A) 3%
B) 6%
C) 12%
D) 20%
E) 24%
Show more

VERITAS PREP OFFICIAL SOLUTION:

The first phase of this problem requires you to determine how many mathematics and chemistry books are even on the shelf. To do so, you have the equations:

m + p + c = 20 (since 4/5 of the 25 spots are full of books)

m = 2p

p = 4 + c

From that, you can use Substitution to get everything down to one variable (and since all three equations include p, that's probably the easiest to work on). If

c = p - 4

m = 2p

p = p

Then (p - 4) + 2p + p = 20, so 4p = 24 and p = 6. That means that there are 12 math, 6 physics, and 2 chemistry books on the shelf.

Here's where this becomes a probability problem: the probability of reading math, then chemistry is (12/20)(2/20), which simplifies to 6/100. That makes B a tempting answer, but keep in mind that Ricardo could read them in the opposite order, chemistry then math. In that case, the probability works out to the same, but since 6% of the time he reads C then M and 6% of the time he reads M then C, then 12% of the time his outcome is that he reads one of each, so the answer is C.
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Mathematics, physics, and chemistry books are stored on a li 28 Jul 2026, 03:37
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So, are we considering the possibility that when Ricardo picked up the second book, that is the one that he is going to take home, if he is not going to pick the first book that he read in the library? It's not explicitly mentioned in the question, but are we looking at that possibility?
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Mathematics, physics, and chemistry books are stored on a li 28 Jul 2026, 12:50
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Hi manupoonia,

Good instinct to question this, because the wording does feel like it should matter. The key is the phrase you may have skimmed past: Ricardo "returns it to the shelf" before the second pick.

That return is what makes both draws happen from the full set of 20 books. That's exactly why Karishma and Bunuel used 20 in the denominator both times - (12/20) for math and (2/20) for chemistry - instead of shrinking the shelf to 19 for the second draw.

So to answer directly: no, the problem is not removing the first book. Because it's replaced, the second draw sees all 20 books again, and yes - he could in principle draw the very same physical book he just read.

But here's why that possibility doesn't affect this answer at all: we're only asked for the probability of one math and one chemistry book. A math book and a chemistry book are, by definition, two different books. So the outcome we're counting can never be "same book twice" - that case only exists for something like "two math books," which isn't what we want. The replacement detail keeps the denominator at 20; the "different book" worry never even comes into play for a math-and-chem result.

A quick way to feel it:
- Roll a die, then roll it again (the die is "returned" each time). P(rolling a 2, then a 5) = (1/6)(1/6). The denominator stays 6 both times - the first roll doesn't shrink the die.
- Getting a 2 and a 5 is automatically two different faces, so "can I repeat?" doesn't change that count.

Same structure here: return = independent draws = denominator stays 20, and a math-and-chem result is always two distinct books anyway.

Answer: C

manupoonia
So, are we considering the possibility that when Ricardo picked up the second book, that is the one that he is going to take home, if he is not going to pick the first book that he read in the library? It's not explicitly mentioned in the question, but are we looking at that possibility?
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Mathematics, physics, and chemistry books are stored on a li 28 Jul 2026, 21:49
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manupoonia
So, are we considering the possibility that when Ricardo picked up the second book, that is the one that he is going to take home, if he is not going to pick the first book that he read in the library? It's not explicitly mentioned in the question, but are we looking at that possibility?
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It is possible that he picked the same book that he read and took it home. But what we need to find is the probability that one book was Math and the other Chem. So we need the probability that the two books were different such that one is Math and the other Chem. So in our possible cases, the option of both books being the same book does not figure.
Hence we simply use 12/20 for Math and after the Math book was kept back, we again had 20 books and hence probability of picking Chem next is 2/20.
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