91Ó°ÊÓ

The table below shows the distribution of books on a bookcase based on whether they are nonfiction or fiction and hardcover or paperback. $$\begin{array}{lccc} & {\text { Format }} & \\ & \text { Hardcover } & \text { Paperback } & \text { Total } \\ \hline \text { Fiction } & 13 & 59 & 72 \\ \text { Nonfiction } & 15 & 8 & 23 \\ \hline \text { Total } & 28 & 67 & 95 \\ \hline\end{array}$$ (a) Find the probability of drawing a hardcover book first then a paperback fiction book second when drawing without replacement. (b) Determine the probability of drawing a fiction book first and then a hardcover book second, when drawing without replacement. (c) Calculate the probability of the scenario in part (b), except this time complete the calculations under the scenario where the first book is placed back on the bookcase before randomly drawing the second book. (d) The final answers to parts (b) and (c) are very similar. Explain why this is the case.

Step by step solution

01

Define Event (a) - Hardcover first, Paperback Fiction second

We need to find the probability of drawing a hardcover book first, followed by a paperback fiction book second, without replacement. First, find the probability of drawing a hardcover book from the total books. Then, find the probability of drawing a paperback fiction book from the remaining books.

02

Probability of a Hardcover Book First

The total number of books is 95, and the number of hardcover books is 28. Thus, the probability of drawing a hardcover book first is \( \frac{28}{95} \).

03

Probability of Paperback Fiction Second

After removing one hardcover book (assuming it isn't a fiction hardcover), 94 books remain. There are 59 paperback fiction books, so the probability of drawing a paperback fiction book second is \( \frac{59}{94} \).

04

Calculate Probability of Combined Event (a)

For the combined event without replacement, multiply the probabilities: \( \frac{28}{95} \times \frac{59}{94} = \frac{1652}{8930} \approx 0.185 \).

05

Define Event (b) - Fiction first, Hardcover second

In this step, calculate the probability of drawing a fiction book first, followed by a hardcover book second, without replacement.

06

Probability of a Fiction Book First

There are 72 fiction books out of a total of 95. Thus, the probability of picking a fiction book first is \( \frac{72}{95} \).

07

Probability of Hardcover Second

After removing one fiction book, 94 books remain. There are 28 hardcover books, so the probability of picking a hardcover book second is \( \frac{28}{94} \).

08

Calculate Probability of Combined Event (b)

Multiply the probabilities for the combined event without replacement: \( \frac{72}{95} \times \frac{28}{94} = \frac{2016}{8930} \approx 0.226 \).

09

Define Event (c) with Replacement - Fiction first, Hardcover second

Here, we calculate the probability of drawing a fiction book and then a hardcover book with replacement.

10

Adjust Probability with Replacement

With replacement, the number of books remains 95 for both draws. Therefore, the probability of a fiction book is \( \frac{72}{95} \), and the hardcover book probability remains \( \frac{28}{95} \). Multiply these probabilities: \( \frac{72}{95} \times \frac{28}{95} = \frac{2016}{9025} \approx 0.223 \).

11

Compare Probabilities from Parts (b) and (c)

The probabilities from parts (b) and (c) are similar because the total number of books is large relative to the changes made by a single draw and because the ratios of fiction and hardcover books are not drastically different.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability without replacement

When drawing objects from a collection without replacing them, it's important to understand that each draw affects the probabilities of subsequent draws. In our book example, when we draw a hardcover book first, it reduces the total number of books remaining. This changes the probability landscape for the second draw:

• The first draw affects the total number; removing a hardcover book means there are only 94 books left for the second draw.
• The specific draw and remaining options make it necessary to adjust probabilities accordingly, as shown through conditional probability calculations.

Remember, without replacement means after each draw, your total and possibly event-specific counts change.

Probability with replacement

With replacement changes the game by resetting the deck after each draw. This means:

• The total number of items (books in this case) remains constant at 95 for both draws.
• Introducing a drawn object back keeps probabilities static between successive draws.

In our book problem: if a fiction book is drawn and then a hardcover book is considered, each draw starts fresh. Hence the straightforward multiplication of independent event probabilities. This reflects in fractional values like this:\[P(\text{Fiction first and Hardcover second with replacement}) = \frac{72}{95} \times \frac{28}{95}\]Thus, you maintain uniform conditions regardless of the outcome of the first draw.

Probability calculation steps

Calculating probabilities involves understanding the parameters and conditions for specific events. Let's break it down:

• Define each event: Here, identify what each draw represents and the respective events for calculation.
• Calculate individual probabilities: Count favorable outcomes over possible (total) outcomes.
• Handle conditions: Check if it's "without replacement" (affects subsequent counts) versus "with replacement" (independent draws).
• Multiply probabilities for combined events: Using the probability multiplication rule for sequential events when their individual probabilities are known.

By following these steps, you can confidently approach similar probability problems.

Book classification probability

Book classification probability looks into defining and understanding how books are categorized based on attributes (like format or genre) and using these classifications to drive probability calculations. For instance:

• In the provided example, books are fiction and nonfiction, hardcover, and paperback. This classification aids in setting up scenarios like drawing specific combinations of books.
• Recognize how these categories influence probability calculations independently (such as having 72 fiction books influencing upfront selections).

By leveraging classification, students can better visualize and execute strategic probability calculations.