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

Explanation:

Step1: Recall dependent events definition

Dependent events: The outcome of one event affects the outcome of the other. So, if we take a book from a shelf and don't replace it (or the second event is related to the same shelf's remaining books), the probability changes.

Step2: Analyze each option

- **Option A**: First, take a fiction book from top shelf (6 fiction, total 10 on top). Then, take a nonfiction from top shelf (4 nonfiction, but after taking one fiction, top shelf has 9 books left). The first event affects the second (since we're taking from the same shelf, reducing the total for the second draw). So these are dependent.
- **Option B**: First, take fiction from top (6 fiction, 10 total top). Then, take fiction from bottom (3 fiction, 8 total bottom). The top shelf draw doesn't affect the bottom shelf's book count (different shelves, independent unless related, but here they are separate). So independent.
- **Option C**: First, take fiction from top (6 fiction, 10 top). Then, take nonfiction from bottom (5 nonfiction, 8 bottom). Different shelves, independent.
- **Option D**: Take nonfiction from bottom (5 nonfiction, 8 bottom), then another nonfiction from bottom. Wait, but the first is bottom nonfiction, second is bottom nonfiction? Wait, no: first is bottom nonfiction (5 nonfiction, 8 total bottom), second is bottom nonfiction. But wait, the first event is taking a nonfiction from bottom, the second is also from bottom. Wait, but the problem says "choosing which 2 books" – wait, no, let's recheck. Wait, Option D: "one of the nonfiction books on the bottom shelf, and a second nonfiction book from the bottom shelf". Wait, but if we take one nonfiction from bottom (5 nonfiction, 8 total bottom), then the second nonfiction from bottom: now, after first draw, bottom shelf has 7 books left, 4 nonfiction. So that's dependent? Wait, no, wait the original problem: top shelf: 6 fiction, 4 nonfiction. Bottom shelf: 3 fiction, 5 nonfiction.

Wait, let's re-express each option:

- **A**: Top shelf fiction (6/10) then top shelf nonfiction (4/9, since one book is removed from top shelf). So dependent, because first event reduces top shelf count, affecting second event's probability.
- **B**: Top shelf fiction (6/10) then bottom shelf fiction (3/8). Independent, because top and bottom are separate, first draw doesn't affect bottom's count.
- **C**: Top shelf fiction (6/10) then bottom shelf nonfiction (5/8). Independent, different shelves.
- **D**: Bottom shelf nonfiction (5/8) then bottom shelf nonfiction (4/7). Wait, but the option says "one of the nonfiction books on the bottom shelf, and a second nonfiction book from the bottom shelf" – but is the first book put back? No, but wait, the problem is about "choosing which 2 books" – but in the context of dependent events, if we are taking two books from the bottom shelf (without replacement), they are dependent. But wait, let's check the original problem again. Wait, the question is "choosing which 2 books describes a pair of dependent events". Wait, maybe I misread Option A. Wait, Option A: "one of the fiction books on the top shelf, and then one of the nonfiction books on the top shelf" – so both from top shelf. So first, take a fiction (6/10), then take a nonfiction (4/9, since top shelf now has 9 books). So dependent. Option D: "one of the nonfiction books on the bottom shelf, and a second nonfiction book from the bottom shelf" – both from bottom shelf. First, 5/8, then 4/7 – dependent. But wait, the top shelf has 10 books (6F,4NF), bottom has 8 (3F,5NF).

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

Dependent events: The first event changes the probability of the second. So, if the two events are from the same shelf (and we don't replace, i.e., the first book is not put back), then they are dependent.

- **Option A**: Both from top shelf. First, take F (6/10), then take NF (4/9) – dependent (since top shelf has 10, then 9 books left).
- **Option B**: Top F (6/10) then bottom F (3/8) – independent (different shelves, top shelf count doesn't affect bottom).
- **Option C**: Top F (6/10) then bottom NF (5/8) – independent (different shelves).
- **Option D**: Bottom NF (5/8) then bottom NF (4/7) – dependent (same shelf, first draw reduces bottom shelf count). But wait, the problem's top shelf has 6F and 4NF (total 10), bottom has 3F and 5NF (total 8).

Wait, but the question is "choosing which 2 books describes a pair of dependent events". Wait, maybe the key is: in Option A, we are taking two books from the same shelf (top), so the first draw affects the second. In Option D, two from bottom. But let's check the answer options again. Wait, maybe I misread Option A: "one of the fiction books on the top shelf, and then one of the nonfiction books on the top shelf" – so first, select a fiction from top (6 options), then a nonfiction from top (4 options). Since we are selecting from the same shelf, the first selection reduces the total number of books on the top shelf, so the probability of the second event is affected. Hence, dependent.

Options B, C: involve different shelves (top then bottom), so the first event doesn't affect the second (since bottom shelf's book count is independent of top shelf's, assuming we don't move books). Option D: two from bottom shelf, but wait, the bottom shelf has 5 nonfiction. Wait, but the option says "one of the nonfiction books on the bottom shelf, and a second nonfiction book from the bottom shelf" – but is the first book being removed? If we are just choosing (not removing), but in probability, dependent events are when the outcome of the first affects the second. So if we are picking two books from the same shelf without replacement, they are dependent. But in Option A, same shelf (top), different types (fiction then nonfiction). So first, take a fiction (6/10), then take a nonfiction (4/9) – dependent. So Option A is dependent.

Answer:

A. one of the fiction books on the top shelf, and then one of the nonfiction books on the top shelf