Probability &
Statistical Foundations

CMSC 178DA - Week 02

Noel Jeffrey Pinton
Department of Computer Science
University of the Philippines Cebu

Learning Objectives

Why Probability Matters

Uncertainty is Everywhere

• Will this customer churn?
• Is this transaction fraudulent?
• Will it rain tomorrow?
• Is this drug effective?

Probability in Data Analytics

Probability is the foundation for everything we'll learn

Set Operations in Probability

Before we combine probabilities, let's review set notation

Basic Probability Rules

Every probability must be between 0 and 1: $0 \leq P(A) \leq 1$

Reading P(A|B) Notation

Before we use this notation, let's understand what it means

Conditional Probability

Independence

Caution: Independence is often assumed but rarely verified!

Introducing Bayes' Theorem

Bayes' Intuition: Updating Beliefs

Bayes gives us a systematic way to update what we believe

Bayes' Theorem

Medical Testing Vocabulary

Before we apply Bayes' theorem, learn these key terms

Bayes' Example: Medical Testing

Most people intuitively guess ~90%. The actual probability is only 8.8%!

Bayes' Calculation

Random Variables

Greek Letter Cheat Sheet

Statistics uses Greek letters as shorthand — here's what they mean

μ and σ describe the population (what we want to know) · $\bar{x}$ comes from our sample (what we can measure)

Expected Value & Variance

Standard Deviation: $\sigma = \sqrt{Var(X)}$ — same units as the data

Example: Is Lotto Worth It?

Let's calculate the expected value of a ₱20 lotto ticket

Wait... that seems profitable? 🤔

Lotto: The Reality Check

Source: PCSO Ultra Lotto 6/58 (official odds)

What is a Probability Distribution?

Different phenomena follow different distributions — choosing the right one is key!

Why the Right Distribution Matters

Always check your data's shape before choosing a distribution

Common Probability Distributions

Different distributions model different types of data: counts, rates, continuous measurements

When to Use: Binomial

Binomial Distribution

from scipy.stats import binom

# P(exactly 5 approvals out of 10, approval rate 30%)
prob = binom.pmf(5, n=10, p=0.3)
print(f"P(X=5) = {prob:.4f}")  # Output: 0.1029

When to Use: Poisson

Poisson Distribution

from scipy.stats import poisson

# P(exactly 25 visits, avg rate 20/hour)
prob = poisson.pmf(25, mu=20)
print(f"P(X=25) = {prob:.4f}")  # Output: 0.0446

The Normal Distribution

The most important distribution in statistics

Normal Distribution

The 68-95-99.7 Rule

• 68% within $\pm 1\sigma$
• 95% within $\pm 2\sigma$
• 99.7% within $\pm 3\sigma$

This rule is used to identify outliers and assess data normality

Central Limit Theorem

As sample size increases: exponential population → normal sample means

CLT Demonstration

# Exponential population (skewed, not normal!)
population = np.random.exponential(scale=2, size=100000)

# Take 1000 samples, compute means → Normal!
sample_means = [np.random.choice(population, 30).mean()
                for _ in range(1000)]

plt.hist(sample_means, bins=30)

Skewed exponential population → Normal distribution of sample means (n=30)

Wrong Distribution = Wrong Conclusions

Choosing the wrong distribution leads to misleading results

Real-World: Fraud Detection

In-Class Exercise

Exercise Answer

Key Takeaways - Probability

Lecture 4

Statistical Inference Essentials

Learning Objectives

From Data to Decisions

Sample vs Population

The fundamental distinction in statistics

Population: PSA 2020 Census (109,035,343)

What is Statistical Inference?

What is Hypothesis Testing?

Just like a jury needs "beyond reasonable doubt," we need a p-value below alpha to reject H₀

The Logic of Hypothesis Testing

Null vs Alternative Hypothesis

We never "prove" H₀ - we either reject it or fail to reject it

Types of Errors in Hypothesis Testing

The Confusion Matrix of Statistical Decisions

Statistical Power = 1 − β  (ability to detect a true effect when it exists)

Remember It: The Pregnancy Test

H₀: "You are NOT pregnant"

Type II errors are often more dangerous — you miss a real condition!

The p-value

Critical Value Tables

Before computers, statisticians looked up critical values in printed tables

Traditional vs Modern Approach

from scipy import stats
t_stat, p_value = stats.ttest_ind(group1, group2)
if p_value < 0.05:
    print("Reject H₀")

Both methods give the same answer — Python just automates the lookup

p-value in Plain Language

How We Got p = 0.02

The coin flip uses the Binomial Distribution

from scipy.stats import binom
p_value = 1 - binom.cdf(8, n=10, p=0.5)
# One-tailed: 0.0107
p_two = 2 * p_value  # Two-tailed: 0.02

Two-tailed p ≈ 0.02 because we'd also reject if we got 9+ tails (equally extreme)

t-test: Comparing Means

t-test Example: Grab Earnings

cebu = [850, 920, 780, 900, 870, 950, 820, 890, 880, 910]
manila = [800, 850, 750, 820, 780, 830, 770, 810, 790, 840]

t_stat, p_value = ttest_ind(cebu, manila)
# Output: t=3.162, p=0.0055 → Significant!

Data simulated based on LTFRB fare structures (hypothetical for illustration)

How t = 3.162 Was Computed

Step-by-step calculation for two-sample t-test

df = 18 → Critical t at α=0.05 is 2.101 → 3.93 > 2.101 → Reject H₀

Chi-Square Test

Chi-Square in Python

# Contingency table: rows=age groups, cols=payment methods
table = np.array([[50, 120, 30],   # Under 30
                   [80, 90, 50],    # 30-50
                   [100, 40, 60]])  # Over 50

chi2, p_value, dof, expected = chi2_contingency(table)
# Output: chi2=62.89, p < 0.0001 → NOT independent!

How χ² = 62.89 Was Computed

Compare Observed (O) vs Expected (E) counts

Introducing Confidence Intervals

Confidence Intervals

Confidence Interval: Range likely to contain the true parameter

$$\bar{x} \pm t_{\alpha/2} \cdot \frac{s}{\sqrt{n}}$$

Correct: 95% of intervals contain the true mean
Wrong: "95% probability the mean is in this interval"

Confidence Interval Example

transactions = [150, 200, 180, 250, 175, 300, 220, 190,
                160, 280, 195, 210, 170, 240, 185]
mean = np.mean(transactions)
se = stats.sem(transactions)  # Standard error
ci = stats.t.interval(0.95, len(transactions)-1, mean, se)
# Output: mean=₱203.67, 95% CI: (₱178.42, ₱228.91)

Simulated data based on GCash transaction patterns

Introducing A/B Testing

A/B Testing Framework

A/B Test Example: Lazada

Multiple Testing Problem

p-value Criticisms

Effect Size: Why It Matters

Cohen's d: Measuring Effect Size

Always report effect size alongside p-values!

In-Class Exercise

Exercise Answer

Key Takeaways - Inference

Lab Preview

Next Week

Week 3: Data Wrangling & Cleaning