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Let's break down the problem step by step and solve it.

### Problem Recap:
We are given the probability density function (PDF) \( f_X(x) \) for the random variable \( X \) representing the time one has to wait for a bus:

\[
f_X(x) = \begin{cases} 
0, & x < 0 \\
\frac{1}{9}(x + 1), & 0 \leq x < 1 \\
\frac{4}{9}\left(x - \frac{1}{2}\right), & 1 \leq x \leq \frac{3}{2} \\
\frac{4}{9}\left(\frac{5}{2} - x\right), & \frac{3}{2} < x \leq 2 \\
\frac{1}{9}(4 - x), & 2 < x < 3 \\
\frac{1}{9}, & 3 \leq x < 6 \\
0, & x \geq 6 
\end{cases}
\]

### Definitions of Events:
- \( A \): One waits between 0 and 2 minutes (inclusive).
- \( B \): One waits between 0 and 3 minutes (inclusive).

We are asked to find \( P(B \mid A) \), the conditional probability of event \( B \) given \( A \).

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

Since \( A \) represents waiting between 0 and 2 minutes and \( B \) represents waiting between 0 and 3 minutes, clearly \( A \cap B = A \) because \( A \) is a subset of \( B \). Therefore:
\[
P(B \mid A) = \frac{P(A)}{P(A)} = 1
\]

Thus, \( P(B \mid A) = 1 \).

### Conclusion:
The conditional probability \( P(B \mid A) \) is 1.

Would you like a more detailed breakdown or any clarifications on this? Here are five related questions to deepen your understanding:

1. How do you compute \( P(A) \) and \( P(B) \) using the PDF?
2. What is the integral of the PDF over specific intervals?
3. How would the problem change if \( B \) were an event involving a different time interval?
4. Can conditional probability be less than 1 in similar setups?
5. How is the expected waiting time computed using the given PDF?

**Tip:** Understanding conditional probability simplifies when events are subsets of one another, as it often results in probabilities of 1 or 0.

The time one has to wait for a bus at a downtown bus stop is observed to be a random phenomenon X with the probability density function provided. The events A and B are defined as follows: A: One waits between 0 to 2 minutes inclusive, and B: One waits between 0 to 3 minutes inclusive. Find P(B | A).

Solve a conditional probability problem where the random variable X represents waiting time for a bus, and the probability density function (PDF) is given. The question involves finding P(B | A) where A is waiting between 0 and 2 minutes and B is waiting between 0 and 3 minutes.

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