select the correct answer.
the top shelf of a bookcase holds 6 fiction and 4 nonfiction books. on the bottom shelf are 3 fiction and 5 nonfiction books.
choosing which 2 books describes a pair of dependent events?

a. one of the nonfiction books on the bottom shelf, and a second nonfiction book from the bottom shelf
b. one of the fiction books on the top shelf, and then one of the nonfiction books on the bottom shelf
c. one of the fiction books on the top shelf, and then a second fiction book from the bottom shelf
d. one of the fiction books on the top shelf, and then one of the nonfiction books on the top shelf

Question

select the correct answer.
the top shelf of a bookcase holds 6 fiction and 4 nonfiction books. on the bottom shelf are 3 fiction and 5 nonfiction books.
choosing which 2 books describes a pair of dependent events?

a. one of the nonfiction books on the bottom shelf, and a second nonfiction book from the bottom shelf
b. one of the fiction books on the top shelf, and then one of the nonfiction books on the bottom shelf
c. one of the fiction books on the top shelf, and then a second fiction book from the bottom shelf
d. one of the fiction books on the top shelf, and then one of the nonfiction books on the top shelf

Accepted Answer

To determine dependent events, we recall that dependent events are those where the outcome of the first event affects the outcome of the second event (usually when we are not replacing the first item chosen). Let's analyze each option:

### Option A:
- First event: Choosing one non - fiction book from the bottom shelf (there are 5 non - fiction books out of $3 + 5=8$ books on the bottom shelf).
- Second event: Choosing a second non - fiction book from the bottom shelf. After choosing the first non - fiction book from the bottom shelf, the number of non - fiction books left on the bottom shelf is $5- 1 = 4$ and the total number of books left on the bottom shelf is $8 - 1=7$. So the outcome of the first event (removing one non - fiction book from the bottom shelf) affects the probability of the second event. These are dependent events. But let's check the other options to be sure.

### Option B:
- First event: Choosing a fiction book from the top shelf (the top shelf has 6 fiction and 4 non - fiction books).
- Second event: Choosing a non - fiction book from the bottom shelf (the bottom shelf has 3 fiction and 5 non - fiction books). The choice of a book from the top shelf does not affect the number of books on the bottom shelf. So these are independent events.

### Option C:
- First event: Choosing a fiction book from the top shelf.
- Second event: Choosing a fiction book from the bottom shelf. The choice of a book from the top shelf does not affect the number of fiction books on the bottom shelf. So these are independent events.

### Option D:
- First event: Choosing a fiction book from the top shelf (top shelf has 6 fiction and 4 non - fiction books).
- Second event: Choosing a non - fiction book from the top shelf. The top shelf has a fixed number of books. But when we choose a fiction book first from the top shelf, we are not removing a non - fiction book. Wait, actually, the top shelf has $6 + 4=10$ books. If we choose a fiction book first (6 choices), the number of non - fiction books on the top shelf is still 4, and the total number of books on the top shelf is still 10? No, wait, if we are choosing books without replacement (which is the usual case for such problems), if we choose a fiction book from the top shelf first, the total number of books on the top shelf for the second choice is $10 - 1=9$, but the number of non - fiction books is still 4. However, in option A, when we choose a non - fiction book from the bottom shelf first, the number of non - fiction books on the bottom shelf for the second choice is reduced. But in option D, the first choice is a fiction book and the second is a non - fiction book from the same (top) shelf. But in option A, both choices are from the same shelf (bottom) and we are choosing the same type (non - fiction) without replacement, so the first choice affects the second. But wait, let's re - examine option D. If we are choosing two books, and the first is a fiction from the top and the second is a non - fiction from the top, when we take the first book (fiction) from the top shelf, the total number of books on the top shelf for the second draw is $6 + 4-1 = 9$, but the number of non - fiction books is still 4. However, in option A, when we take the first non - fiction from the bottom shelf (5 non - fiction out of 8), then the second non - fiction from the bottom shelf has 4 non - fiction out of 7. So the probability of the second event in option A is affected by the first event. But wait, let's go back.

Wait, the key for dependent events is that the outcome of the first event changes the probability of the second event. In option A, when we pick a non - fiction book from the bottom shelf first, the number of non - fiction books available for the second pick from the bottom shelf decreases, so the probability of the second non - fiction book from the bottom shelf is affected. In option D, when we pick a fiction book from the top shelf first, the number of non - fiction books on the top shelf is still 4, and the total number of books on the top shelf is 9 (if we don't replace). But the type of the first book (fiction) doesn't affect the number of non - fiction books. Wait, no, the first book is fiction, the second is non - fiction. The first event is choosing a fiction book, the second is choosing a non - fiction book from the same shelf. The total number of books on the top shelf for the second choice is $10 - 1=9$, but the number of non - fiction books is 4. So the probability of the second event is $\frac{4}{9}$ regardless of the first event? No, wait, no. The first event is choosing a fiction book (6 choices), the second is choosing a non - fiction book (4 choices). If we consider the probability, the probability of the first event is $\frac{6}{10}$, and the probability of the second event given the first event is $\frac{4}{9}$. But in option A, the probability of the first event (non - fiction from bottom) is $\frac{5}{8}$, and the probability of the second event (non - fiction from bottom) given the first is $\frac{4}{7}$. In option D, the two events are from the same shelf, but the first is fiction and the second is non - fiction. However, in option A, both events are from the same shelf and same category (non - fiction) without replacement, so the first event affects the second. But wait, let's check the original problem again.

Wait, maybe I made a mistake. Let's re - analyze:

Dependent events: $P(A\cap B)=P(A)\times P(B|A)$, where $P(B|A)
eq P(B)$

For option A:

Let $A$ be the event of choosing a non - fiction book from the bottom shelf. $P(A)=\frac{5}{8}$

Let $B$ be the event of choosing a non - fiction book from the bottom shelf. $P(B|A)=\frac{4}{7}$ (since after choosing one non - fiction book from the bottom shelf, there are 4 non - fiction left out of 7 total on the bottom shelf)

$P(B)=\frac{5}{8}$ (if we didn't choose $A$ first). Since $P(B|A)
eq P(B)$, $A$ and $B$ are dependent.

For option D:

Let $A$ be the event of choosing a fiction book from the top shelf. $P(A)=\frac{6}{10}$

Let $B$ be the event of choosing a non - fiction book from the top shelf. $P(B)=\frac{4}{10}$

$P(B|A)=\frac{4}{9}$ (since after choosing a fiction book from the top shelf, there are 9 books left, 4 non - fiction)

$P(B)=\frac{4}{10}
eq\frac{4}{9}$, so $A$ and $B$ are also dependent? Wait, this is confusing. Wait, maybe the problem is about whether the two events are from the same set of books (same shelf) and we are not replacing.

In option A, both events are from the bottom shelf, same type (non - fiction), and we are choosing without replacement (since we are choosing 2 books). So the first choice affects the second.

In option D, both events are from the top shelf, different types (fiction and non - fiction), and we are choosing without replacement. So the first choice (fiction) affects the total number of books on the top shelf for the second choice.

But let's check the answer options again. The key is: Dependent events occur when the outcome of the first event affects the outcome of the second event.

In option A: Choosing a non - fiction from the bottom shelf first, then a second non - fiction from the bottom shelf. Since we are choosing from the same shelf and same type without replacement, the first choice reduces the number of non - fiction books on the bottom shelf, so the probability of the second event is affected.

In option B: The two events are from different shelves (top and bottom), so the choice from the top doesn't affect the bottom.

In option C: The two events are from different shelves (top and bottom), so the choice from the top doesn't affect the bottom.

In option D: The two events are from the same shelf (top), but one is fiction and one is non - fiction. Wait, if we choose a fiction book from the top shelf first, the number of non - fiction books on the top shelf is still 4, and the total number of books on the top shelf is 9 (if we don't replace). But the first event is choosing a fiction book, the second is choosing a non - fiction book. The first event doesn't affect the number of non - fiction books. Wait, no, the total number of books on the top shelf is 10. If we take a fiction book out, the total number of books for the second choice is 9, but the number of non - fiction books is still 4. So the probability of the second event (non - fiction from top) given the first event (fiction from top) is $\frac{4}{9}$, and the probability of the second event without the first event (if we were choosing a non - fiction book first) would be $\frac{4}{10}$. Wait, no, the second event is non - fiction from top, regardless of the first event. Wait, maybe I messed up.

Wait, let's think of it as:

- Dependent events: When the two events are related such that the occurrence of the first changes the sample space for the second.

In option A: The sample space for the second event is the bottom shelf with one non - fiction book removed. So the sample space changes.

In option D: The sample space for the second event is the top shelf with one fiction book removed. But the second event is about non - fiction books. The number of non - fiction books on the top shelf is still 4, but the total number of books is 9. So the probability of choosing a non - fiction book is $\frac{4}{9}$ after choosing a fiction book, and if we had chosen a non - fiction book first, the probability of choosing a non - fiction book (but that's a different event). Wait, the event in option D is first fiction, then non - fiction from the top shelf. So the first event is choosing a fiction book (6 out of 10), the second is choosing a non - fiction book (4 out of 9). The first event affects the total number of books on the top shelf, but not the number of non - fiction books. However, in option A, the first event (non - fiction from bottom) affects the number of non - fiction books on the bottom shelf for the second event.

But let's check the answer. The correct answer is A. Because in option A, we are choosing two non - fiction books from the bottom shelf without replacement. So the first choice affects the second choice. In option D, we are choosing a fiction and a non - fiction from the top shelf. If we choose a fiction first, the number of non - fiction books on the top shelf is still 4, and the total number of books on the top shelf is 9. But the key is that in option A, the two events are from the same set of books (bottom shelf non - fiction) and we are not replacing, so the first event affects the second. In option D, the two events are from the same shelf but different types, and the first event (fiction) doesn't affect the number of non - fiction books. Wait, no, the total number of books on the top shelf is reduced by 1, but the number of non - fiction books is not. So the probability of the second event (non - fiction from top) is $\frac{4}{9}$ after the first event (fiction from top), and if we had chosen a non - fiction book first, the probability of a non - fiction book would be $\frac{3}{9}$, but that's a different event. The event in option D is specifically non - fiction after fiction, so the first event (fiction) doesn't affect the number of non - fiction books, only the total number of books. But in option A, the first event (non - fiction) affects the number of non - fiction books available for the second event.

So the correct answer is A.

# Answer:
A. one of the nonfiction books on the bottom shelf, and a second nonfiction book from the bottom shelf

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Option A:

Option B:

Option C:

Option D:

Answer: