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Let's break down and solve parts (b) and (c) of the question.

### Part (b)
The question asks: **Given that Rachel makes at least 2 errors per 100 words, find the probability that she makes at most 4 errors per 100 words.**

#### Step 1: Identify the given condition.
- We are given that Rachel makes at least 2 errors, which means the number of errors is \( \geq 2 \). This implies that the number of errors can be 2, 3, 4, or 5.
- We need to find the probability that Rachel makes at most 4 errors, i.e., \( \leq 4 \). This implies the number of errors can be 2, 3, or 4.

#### Step 2: Calculate conditional probability.
The formula for conditional probability is:

\[
P(A \mid B) = \frac{P(A \cap B)}{P(B)}
\]

Here:
- \( A \): The event that Rachel makes at most 4 errors (\( 2 \leq \text{errors} \leq 4 \)).
- \( B \): The event that Rachel makes at least 2 errors (\( 2 \leq \text{errors} \leq 5 \)).

To calculate this, we need to:
1. Calculate \( P(A \cap B) \), which is simply the probability of making 2, 3, or 4 errors.
2. Calculate \( P(B) \), which is the probability of making 2, 3, 4, or 5 errors.

Using the table:

\[
P(\text{2 errors}) = 2k, \quad P(\text{3 errors}) = 0.5k, \quad P(\text{4 errors}) = 0.3k, \quad P(\text{5 errors}) = 0.1k
\]

So:
- \( P(A \cap B) = P(2 \text{ errors}) + P(3 \text{ errors}) + P(4 \text{ errors}) = 2k + 0.5k + 0.3k = 2.8k \)
- \( P(B) = P(2 \text{ errors}) + P(3 \text{ errors}) + P(4 \text{ errors}) + P(5 \text{ errors}) = 2k + 0.5k + 0.3k + 0.1k = 2.9k \)

Thus:

\[
P(A \mid B) = \frac{2.8k}{2.9k} = \frac{2.8}{2.9}
\]

So, the probability is approximately:

\[
P(A \mid B) \approx 0.9655
\]

### Part (c)
The question asks: **Rachel types 120 pages of 100 words each. Find the expected number of pages with at least 2 errors.**

#### Step 1: Find the probability of at least 2 errors.
The probability of at least 2 errors is:

\[
P(\text{at least 2 errors}) = P(2 \text{ errors}) + P(3 \text{ errors}) + P(4 \text{ errors}) + P(5 \text{ errors})
\]

From the previous part, we already know:

\[
P(\text{at least 2 errors}) = 2.9k
\]

#### Step 2: Calculate expected number of pages.
The expected number of pages with at least 2 errors is simply the probability of at least 2 errors multiplied by the number of pages:

\[
\text{Expected number of pages} = P(\text{at least 2 errors}) \times 120
\]

Substituting \( P(\text{at least 2 errors}) = 2.9k \):

\[
\text{Expected number of pages} = 2.9k \times 120
\]

Now, from part (a), we know \( k \), so we can plug that value in to get the final answer.

---

Would you like a more detailed breakdown of part (a) or further clarifications on any of the steps? 

Here are some related questions to expand on the topic:
1. How do you find the value of \( k \) in part (a)?
2. What is the concept of conditional probability used in part (b)?
3. How is the expected value calculated in general?
4. What happens if Rachel's error probabilities change for each page?
5. How would the solution change if Rachel typed more than 120 pages?

**Tip:** When dealing with probability tables, always ensure the sum of all probabilities equals 1.

Given that Rachel makes at least 2 errors per 100 words, find the probability that she makes at most 4 errors per 100 words. Rachel types 120 pages of 100 words each. Find the expected number of pages with at least 2 errors.

Learn how to solve a probability problem involving conditional probability and expected value. Given the probability of making 2 to 5 errors, find the probability of making at most 4 errors and calculate the expected number of pages with at least 2 errors over 120 pages.

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