Danny and Owen

*Later at the end of practice, the three of us are in the practice room, the room we usually use for dance choreography practice. My hair is slightly messy.* "Ah..it's hot.." *I said in a soft voice, pouting a little. The three of us were tired after practice, and all of us were sweating. Even though we are idols, we are still human. I was lying on the floor of the practice room. There were several staff there, one of whom was the one who trained our dance choreography and our singing.*

To determine the probability that a smoker has lung cancer (P(Cancer|Smoker)), we use Bayes' theorem: **Bayes' Theorem:** \[ P(\text{Cancer}|\text{Smoker}) = \frac{P(\text{Smoker}|\text{Cancer}) \times P(\text{Cancer})}{P(\text{Smoker})} \] **Given:** - \( P(\text{Smoker}|\text{Cancer}) = 85\% = 0.85 \) - \( P(\text{Cancer}) = 0.12\% = 0.0012 \) - \( P(\text{Smoker}|\text{No Cancer}) = 15\% = 0.15 \) - \( P(\text{No Cancer}) = 1 - 0.0012 = 0.9988 \) **Calculate \( P(\text{Smoker}) \):** \[ \begin{align*} P(\text{Smoker}) &= P(\text{Smoker}|\text{Cancer}) \times P(\text{Cancer}) + P(\text{Smoker}|\text{No Cancer}) \times P(\text{No Cancer}) \\ &= (0.85 \times 0.0012) + (0.15 \times 0.9988) \\ &= 0.00102 + 0.14982 \\ &= 0.15084 \end{align*} \] **Apply Bayes' Theorem:** \[ \begin{align*} P(\text{Cancer}|\text{Smoker}) &= \frac{0.85 \times 0.0012}{0.15084} \\ &= \frac{0.00102}{0.15084} \\ &\approx 0.00676 \quad (\approx 0.676\%) \end{align*} \] **Answer:** The probability that a smoker has lung cancer is approximately **0.68%**.

To find the path of a Python module, you can use several methods depending on your needs. Below are the most common approaches: --- ### **1. Using the `__file__` Attribute** Most modules have a `__file__` attribute that stores their file path. **Example**: ```python import os print(os.__file__) # Output: Path to the `os` module (e.g., '/path/to/os.py') ``` **Caveats**: - **Built-in modules** (e.g., `sys`, `math`) may not have this attribute. - For **packages**, this points to the `__init__.py` file. --- ### **2. Using the `inspect` Module** The `inspect.getfile()` function reliably retrieves the module path, even for nested packages. **Example**: ```python import inspect import pandas # Third-party module print(inspect.getfile(pandas)) # Output: Path to pandas/__init__.py ``` **Caveats**: - Raises `TypeError` for built-in modules (e.g., `sys`). --- ### **3. Using `importlib` (Python 3.4+)** The `importlib` module provides a programmatic way to locate modules. **Example**: ```python import importlib.util spec = importlib.util.find_spec("requests") if spec: print(spec.origin) # Output: Path to `requests/__init__.py` ``` **Caveats**: - Returns `None` for built-in modules. --- ### **4. Command Line Method** Use a one-liner in the terminal to print the path: ```bash python -c "import os; print(os.__file__)" ``` --- ### **Handling Built-In Modules** Built-in modules (e.g., `sys`, `math`) are compiled into the Python interpreter and lack a file path. **Example**: ```python import sys print(sys) # Output: <module 'sys' (built-in)> ``` --- ### **Summary Table** | Method | Example | Works For | Fails For | |----------------------|----------------------------------|-------------------------|-------------------------| | `module.__file__` | `os.__file__` | Most modules | Built-in modules | | `inspect.getfile()` | `inspect.getfile(pandas)` | Most modules | Built-in modules | | `importlib` | `importlib.util.find_spec(...)` | Third-party modules | Built-in modules | | Command Line | `python -c "..."` | Quick checks | Built-in modules | --- ### **Key Notes**: - Use `__file__` for simple cases. - Prefer `inspect` for nested packages. - Use `importlib` for dynamic imports. - Built-in modules (e.g., `sys`) have no file path.

*The crowd roars with excitement, a mix of anticipation and anxiety. The air is thick with tension as the host hands the microphone back to the old man. He clears his throat, scanning the crowd with a knowing smile.* "Before we announce the winner, let's take a moment to recognize the outstanding performances from all the groups tonight. Their dedication and talent have truly impressed us all." *The crowd cheers, a mix of excitement and relief. The old man pauses, glancing at the stage.* "And now, without further ado, the moment we've all been waiting for. Ladies and gentlemen, the winner of this year's Best Music Group Award is..." *He trails off, his gaze drifting toward the three of you.*

To integrate \( f(x) = x^2 \sqrt{1 + x^3} \), we use substitution: 1. **Let \( u = 1 + x^3 \)**. Then, \( \frac{du}{dx} = 3x^2 \) → \( du = 3x^2 dx \) → \( x^2 dx = \frac{du}{3} \). 2. **Rewrite the integral** in terms of \( u \): \[ \int x^2 \sqrt{1 + x^3} \, dx = \int \sqrt{u} \cdot \frac{du}{3} = \frac{1}{3} \int u^{1/2} \, du \] 3. **Integrate** \( u^{1/2} \): \[ \frac{1}{3} \cdot \frac{2}{3} u^{3/2} + C = \frac{2}{9} u^{3/2} + C \] 4. **Substitute back** \( u = 1 + x^3 \): \[ \frac{2}{9} (1 + x^3)^{3/2} + C \] **Final Answer:** \[ \boxed{\frac{2}{9} (1 + x^3)^{3/2} + C} \]