Lesson 3 — Probability Foundations in Context (with Simulation)

Why this matters (motivation)¶

In business/economics, you rarely get certainty. You get risk:

• Will demand exceed capacity?

• Will a customer churn?

• Will a screening rule flag too many false positives?

• How much revenue variability should we expect next month?

Probability gives a disciplined way to think about these questions.

Probability as a language for uncertainty¶

Frequentist intuition (what we use today)¶

If you repeat a process many times (real or simulated), probability is the long-run proportion of outcomes.

Random variables (intuitive definitions)¶

Two core quantities¶

• Expected value: the long-run average (“typical level”)

• Variance / SD: the typical spread (“how uncertain”)

Common business distributions (light intuition)¶

Bernoulli (yes/no outcomes)¶

• Example: customer churns (yes/no), email clicked (yes/no)

Binomial (number of successes out of n)¶

• Example: out of 100 customers contacted, how many convert?

Arrivals (count per time, intuition only)¶

• Example: customer arrivals per hour, website requests per minute

• We often model these as a “count process” with variability.

Simulation (Monte Carlo): your practical workhorse¶

Why simulation is powerful¶

• Works even when formulas are messy

• Makes assumptions explicit

• Helps build intuition quickly

Mini case 1: Capacity planning (arrivals)¶

Scenario: A café has capacity for 40 customers per hour.
You want to estimate: $P(\text{arrivals} > 40)$.

Steps:

1. Choose a simple model for arrivals (we’ll start with a reasonable assumption).

2. Simulate many hours.

3. Count the share of hours where arrivals exceed 40.

Mini case 2: Marketing conversion (binomial)¶

Scenario: Conversion rate is about 5%. You contact 200 customers.
What is:

• expected conversions?

• probability conversions are below 5 (a disappointing campaign)?

• how variable is the outcome?

This is a binomial-style situation, and simulation gives quick answers.

Mini-lab (Google Colab)¶

In-class tasks (checkpoints)

1. Simulate 10,000 Bernoulli trials and estimate a probability.

2. Simulate a binomial process (conversions out of $n$) and summarize the distribution.

3. Compute:

   • an estimated probability of a “bad outcome” (e.g., conversions ≤ threshold),

   • an expected value,

   • and a simple uncertainty summary (e.g., 5th–95th percentile).

4. Write a short interpretation in business language:

   • “What is likely?”

   • “What is risky?”

   • “What action would you take?”

Submission

• Colab link (view permission) or PDF export.

AI check (responsible use for probability work)¶

Good prompt examples

• “Write Python code to simulate conversions with p=0.05 for n=200 repeated 10,000 times.”

• “How do I compute the probability conversions ≤ 5 from simulated results?”

Bad prompt example

• “Give me the final answer and interpretation without showing calculations.”

Review questions (quiz / reflection)¶

1. What is the difference between expected value and most likely outcome?

2. Why can simulation be useful even when a formula exists?

3. In a conversion campaign, which risk matters more: unusually low conversions or unusually high conversions? Why?