Welcome to the Math Cheat-Sheet — your one-stop shop for remembering just enough math to sound impressive in meetings and actually understand your ML models.

You’ve survived symbols, matrices, derivatives, and probabilities. Now it’s time to stitch them all together into practical business cases.

🧮 1. Quick Recap: What We’ve Learned¶

💼 2. Worked Example: Linear Regression = Business Forecasting¶

Goal: Predict sales from ad spend.

[ y = β₀ + β₁x ]

Example: [ y = 5000 + 3x ]

If you spend $1,000 on ads: [ y = 5000 + 3(1000) = 8000 ]

💬 “For every $1 spent on ads, we get$3 in sales uplift.” ROI just became an equation!

🧩 Try It Yourself¶

# 👇 Run this in JupyterLite or Colab
import numpy as np

# Ad spend in $
x = np.array([0, 500, 1000, 1500])
# Sales in $
y = 5000 + 3*x

for spend, sale in zip(x, y):
    print(f"Ad spend: ${spend:>4} → Predicted sales: ${sale}")

📉 3. Worked Example: Derivative = Sensitivity Analysis¶

Say your revenue depends on price: [ R(p) = p \times (100 - 2p) ]

Derivative: [ \frac{dR}{dp} = 100 - 4p ]

Set derivative = 0 to find the optimal price: [ 100 - 4p = 0 \Rightarrow p = 25 ]

✅ Optimal price: $25 per unit ✅ Maximum revenue: ( R(25) = 25 × (100 - 50) = 1250 )

💬 “You just did profit optimization — without Excel!”

🧩 Try It Yourself¶

import sympy as sp

p = sp.Symbol('p')
R = p * (100 - 2*p)
optimal_price = sp.solve(sp.diff(R, p), p)[0]
max_revenue = R.subs(p, optimal_price)
float(optimal_price), float(max_revenue)

🎲 4. Worked Example: Probability = Customer Retention¶

Scenario: 20% of customers are at high churn risk. If they call support, churn probability drops to 5%. 60% of at-risk customers actually call support.

What’s the overall churn rate?

Using total probability: [ P(\text{Churn}) = P(\text{Support})P(\text{Churn|Support}) + P(\text{No Support})P(\text{Churn|No Support}) ] [ = (0.6)(0.05) + (0.4)(0.20) = 0.03 + 0.08 = 0.11 ]

✅ Overall churn = 11%

💬 “A 9% improvement just by nudging customers to call support.” Probability = revenue protection math!

🧩 Try It Yourself¶

p_support = 0.6
p_churn_support = 0.05
p_churn_no_support = 0.20

p_churn = (p_support*p_churn_support) + ((1-p_support)*p_churn_no_support)
print(f"Overall churn probability: {p_churn:.2%}")

🧠 5. Worked Example: Expected Value = Smart Business Betting¶

Scenario: A marketing campaign has 3 possible outcomes:

[ E[X] = (0.2)(50000) + (0.5)(10000) + (0.3)(-5000) = 10,000 + 5,000 - 1,500 = 13,500 ]

✅ Expected profit = $13,500

💬 “Even risky campaigns can be smart bets if the math checks out.”

🧩 Try It Yourself¶

outcomes = np.array([50000, 10000, -5000])
probs = np.array([0.2, 0.5, 0.3])

expected_value = np.sum(outcomes * probs)
print(f"Expected campaign profit: ${expected_value:,.0f}")

🧩 Bonus: Variance = Business Uncertainty¶

Let’s see how volatile that campaign could be:

[ Var(X) = E[X^2] - (E[X])^2 ]

variance = np.sum((outcomes**2) * probs) - expected_value**2
std_dev = np.sqrt(variance)
print(f"Variance: {variance:,.0f}, Std Dev: ${std_dev:,.0f}")

💬 “Variance is your CFO’s favorite anxiety metric.”

🧭 6. Quick Business-Math Reference¶

🧩 Practice Corner: “Business Math Speed Round”¶

Fill in the blanks:

✅ Hint: You’ve seen all of these above. Try coding them in your Colab workspace!

🚀 Summary¶

You can now:

• Speak math and business fluently 🧮💼

• Translate formulas into insights

• Run tiny Python tests to verify your intuition

• Impress both data scientists and executives

🎉 Congratulations! You’ve officially completed the Math Foundations of Machine Learning for Business. You now possess the power to decode formulas like they’re budget reports.

🔜 Next Up¶

👉 Head to Data Handling & Visualization to turn theory into real business intelligence.