🎯 Why Probability Matters

Probability is everywhere in AI:

• Model uncertainty
• Classification confidence
• Loss functions
• Bayesian thinking

Interviewers want intuition, not formulas only.

🧮 Basic Definitions

Probability

Example:

P(head) = 1 / 2

🎲 Conditional Probability

Example:

Given it rains, what’s the chance traffic is heavy?

🔁 Independence

Two events are independent if: [ P(A \cap B) = P(A)P(B) ]

Interview trick question ⚠️ 👉 Independence ≠ no correlation always

🧠 Bayes’ Theorem (VERY IMPORTANT)

Used in:

• Spam detection
• Medical diagnosis
• Bayesian ML

🧪 Interview Question

Why Naive Bayes works despite false independence assumptions?

✅ Because it reduces variance and scales well.

🏁 Takeaway

Probability gives AI confidence, not just predictions.

📘 Gradient Descent (Interview Intuition)

🎯 Problem

How do machines learn?

By minimizing a loss function.

📉 Loss Function

🧠 Gradient Descent Idea

Walk downhill in the direction of steepest descent

🧑‍💻 Python Example

def gradient_descent(w, x, y, lr=0.01):
    grad = -2 * x * (y - w * x)
    return w - lr * grad

⏱️ Variants Interviewers Love

🚨 Interview Trap

Why not use a large learning rate?

❌ Overshooting ❌ Divergence

🏁 Takeaway

Optimization is the engine of AI.

📘 NumPy for Coding Interviews

🎯 Why NumPy?

Interviewers expect:

• Vectorized thinking
• Clean math code
• Speed awareness

🧮 Arrays vs Lists

import numpy as np

a = np.array([1, 2, 3])

Why better?

• Faster
• Cleaner
• Mathematical

➕ Vector Operations

a + b        # element-wise
a * b
np.dot(a,b) # dot product

🧱 Matrix Multiplication

A @ B

Equivalent to:

np.matmul(A, B)

📏 Broadcasting (Interview Favorite)

A + 1

Adds 1 to every element.

🏁 Takeaway

NumPy replaces loops with math.

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# Time vs Space Trade-offs

## 🎯 Interview Core Concept

You can optimize:
- Time
- Space

Rarely both.

---

## 🧩 Example: Duplicate Detection

### Less Space, More Time

```python
for i in range(n):
    for j in range(i+1, n):
        if nums[i] == nums[j]:
            return True

⏱️ O(n²), 💾 O(1)

More Space, Less Time

seen = set()
for v in nums:
    if v in seen:
        return True
    seen.add(v)

⏱️ O(n), 💾 O(n)

🧠 Interview Question

Which would you choose?

Correct answer:

Depends on constraints.

🏁 Takeaway

Strong engineers trade resources intentionally.