WHY THIS MATTERS

Start with Data Science: Where Does Probability Enter?

CMSC 320 • Basic Probability & Distributions

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Why Does a Data Scientist Need Probability & Distributions?

• Fraud model: P(Fraud | transaction features) = 0.82
• Churn model: P(Churn | customer history) = 0.31
• A/B test: 12.4% conversion vs. 10.8%
• Service metric: 95th-percentile latency = 420 ms
• Housing data: mean price = $510K, median price = $390K

TOPICS

Today’s Roadmap

CMSC 320 • Basic Probability & Distributions

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Events &�probability

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Conditional�reasoning

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Bayes &�total probability

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Independence &�expected value

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Common�distributions

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Normal, Z�& CLT

Goal: connect each mathematical idea to a data-science decision.

Textbook reference: Chapter 6

DS Scenario 1: Which Customers Are Likely to Churn?

Dataset: 10,000 customers

Features: tenure, monthly charge, contract type, support calls

Target: churn = Yes / No

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Overall churn rate = 30%, but customer groups may behave differently.

We want P(Churn | observed customer information), not only P(Churn).

Needed concepts → events, probability axioms, conditional probability, independence, Bayes' theorem.

CORE CONCEPTS

Probability Basics: Outcomes → Events → Probability

CMSC 320 • Basic Probability & Distributions

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Sample space Ω

All possible outcomes��Die: {1, 2, 3, 4, 5, 6}

Event A

A subset of outcomes of interest��A = “roll an even number”

Probability P(A)

Likelihood of the event��P(A) = 3/6 = 0.5

0 ≤ P(A) ≤ 1

Data science translation: define the population → define the event → quantify uncertainty.

CORE CONCEPTS

From Events to Random Variables

CMSC 320 • Basic Probability & Distributions

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A random variable assigns a number to an uncertain outcome.

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X = sum of two dice

Distribution

Each possible X value has an associated probability.��P(X=7)=6/36

Focus on the pattern across outcomes—not one roll.

CONDITIONAL REASONING

Conditional Probability: Information Changes the Sample Space

CMSC 320 • Basic Probability & Distributions

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P(A | B) = P(A ∩ B) / P(B)

Marble example

10 marbles total�5 red • 3 of the red are shiny��Given RED → only 5 marbles remain relevant

3 shiny red

P(shiny | red) = 3/5 = 0.60

Most real-world predictions are conditional.

What is the probability that a marble is shiny, given that it is red?

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Conditional probability from a table

Always inspect the denominator

Observed users

Churn

Stay

Total

Premium

80

120

200

Standard

240

560

800

Total

320

680

1000

Question 1: Among Premium users, what proportion churned?

P(Churn | Premium) = 80/200 = 0.40

Question 2: Among users who churned, what proportion were Premium users?

P(Premium | Churn) = 80/320 = 0.25

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Important

P(A|B) and P(B|A) are generally different.

CMSC320 • Introduction to Data Science

Basic Probability and Distributions — Part 1

RELATIONSHIPS

Independence: Does Knowing B Change A?

CMSC 320 • Basic Probability & Distributions

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Independent ⇔ P(A | B) = P(A)

Equivalent: P(A ∩ B) = P(A)P(B)

Independent

Two separate fair coin tosses

Not independent

Rain and umbrella use

Modeling question

Does one feature add information about another?

Independence is an assumption to check—not simply assume.

In data science, independence assumptions appear in models and statistical tests

Knowing B does not change the probability of A

RELATIONSHIPS

Conditional Independence: Context Can Remove a Relationship

CMSC 320 • Basic Probability & Distributions

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Rain

Dog barks

Cat runs

Given Dog barks: P(Cat runs | Rain, Dog barks) = P(Cat runs | Dog barks)

Once the middle variable is known, Rain adds no extra information about Cat runs.

Why DS cares: conditional independence simplifies models such as Naive Bayes.

CONDITIONAL REASONING

Bayes’ Rule: Update a Belief with Evidence

P(A | B)=P(B | A)•P(A)/P(B)

Prior

P(A)

What we believed before

Likelihood

P(B | A)

How compatible is evidence?

Evidence

P(B)

How common is the evidence?

Posterior

P(A | B)

Updated belief

Example:update P(spam) after seeing the word “discount”.

CMSC 320 • Basic Probability & Distributions

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Example: Picnic Day

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What is the chance of rain during the day?

Scenario: Find the chance of rain given that the morning is cloudy.

1. P(Rain) = 10%

Only 3 out of 30 days are rainy.

2. P(Cloud | Rain) = 50%

50% of rainy days start off cloudy.

3. P(Cloud) = 40%

40% of all days start off cloudy.

Applying Bayes' Theorem:

Only a 12.5% chance of rain.

Not too bad, let's have a picnic! 🧺☀️

PROBABILITY LAWS

Law of Total Probability: Add Across Possible Causes

When an event can happen through several non-overlapping cases:

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Example: Two Bags of Marbles

Law of Total Probability: back to the example

Sample Space S: The scenario of choosing a bag and drawing a marble.

1. Mutually Exclusive Events B1​ and B2​:
• B1​: Selecting Bag 1.
• B2​: Selecting Bag 2.

These events are mutually exclusive because you can only choose one bag at a time.

1. Event A: Drawing a black marble.

According to the law of total probability, the probability of drawing a black marble (event A) is:

P(A)=P(B1)⋅P(A | B1)+P(B2)⋅P(A |B2)P(A)

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Try Yourself

Scenario: Imagine you are organizing a charity event, and there are three possible venues (A, B, and C) where you can hold the event. The probability of each venue being available on a given day is as follows:

Venue A: 40% chance of being available.

Venue B: 30% chance of being available.

Venue C: 30% chance of being available.

You also know that if Venue A is available, there's a 70% chance of raising a large amount of money for the charity, while if Venue B or C is available, there's a 50% chance of raising a large amount of money.

What is the overall probability L of raising a large amount of money at the charity event?

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DECISION MAKING

Expected Value: Average Behavior Over Repetition

CMSC 320 • Basic Probability & Distributions

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E[X] = Σ x · P(X=x)

What it means

Long-run average outcome�Not a guaranteed single outcome�Can represent reward, cost, clicks, loss, or risk

20%

$0

30%

$10

40%

$20

10%

$10

EV = $12 per spin

In ML, optimization often means minimizing expected loss.

Expected Value (Average Behavior)

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Probability Distributions and Their Types

Distributions describe how values are spread.

1(a,b). Bernoulli and Binomial Distribution

Bernoulli Distribution

Binary Outcome: Success (1) or Failure (0)

• Models a single trial
• Success probability = p
• P(X = 1) = p
• P(X = 0) = 1 − p
• Example: Single coin flip, spam vs. not spam

Binomial Distribution

Number of successes (k) in n independent Bernoulli trials.

X ~ Binomial(n, p)

Example: Heads in 10 flips, quiz scores

Key Idea: Binomial = sum of independent Bernoulli trials

Worked Example

Tossing a fair coin 10 times:

n = 10, p = 0.5

1a. Poisson Distribution

What it Models

Counts of events occurring in a fixed time or space interval, where events occur randomly at a constant average rate.

Key Assumptions

• Events occur independently of one another
• Average rate λ remains constant
• Simultaneous events cannot occur at exact same instant

Probability Mass Function (PMF)

Where:

• P(X=k): Probability of observing exactly k events
• λ: Average rate parameter per interval
• e: Euler's constant (≈ 2.71828)

X ~ Poisson(λ)

Distribution Shape

Right-skewed for small values of λ.

Becomes increasingly symmetric as λ increases.

Worked Example

Emails arrive at an average rate of 3 per hour. What is the probability of receiving exactly 2 emails in 1 hour?

Given Parameters:

• λ = 3 events/hour | t = 1 hour | k = 2 emails

Zero-Inflated Poisson (ZIP) Distribution

Used for count data with more zeros than expected.

Components

• Poisson component: models the count process
• Inflation component: models excess zeros (e.g., Bernoulli)

Why ZIP?

• Standard Poisson underestimates zeros
• ZIP captures structural zeros + counts�

Examples

• Insurance claims
• Hospital visits
• Product defects

Oftentimes, poisson distributions can have a spike at zero.

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2a. The Uniform Distribution

Key Characteristics

A continuous or discrete probability distribution where all outcomes within a specified boundary are equally likely.

• Equally Likely: Every outcome has the exact same probability of occurrence.
• Fixed Interval: Values strictly occur within a defined minimum and maximum range.

Classic Example: Fair Six-Sided Die

Rolling a standard, balanced die gives 6 discrete outcomes {1, 2, 3, 4, 5, 6}.

Probability Mass Function:

• P(X = k) = 1/n = 1/6 ≈ 16.67% for each face
• Flat probability density across all valid outcomes.

Parameters of the Normal Distribution: μ & σ

Mean (μ)

Controls the position of the curve

• Shifts the curve left or right
• Controls the center of the distribution

Standard Deviation (σ)

Controls the shape and dispersion of the curve

• Controls the spread of the curve
• Smaller σ → narrower, taller
• Larger σ → wider, shorter

Problem Solving: Finding percentages

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Example 1 (Solved)

The travel time between two regional cities is approximately normally distributed with a mean of 70 minutes and a standard deviation of 2 minutes.

Q: What is the percentage of travel times that are between 66 minutes and 72 minutes?

Try Yourself

The volume of soup served via machine is normally distributed with a mean of 240 mL and a standard deviation of 5 mL. A store serves 160 cups of soup.

Q: What number of these cups of soup are expected to contain less than 230 ml?

Solution & Parameters

• Mean (μ) = 70 minutes
• Standard Dev (σ) = 2 minutes
• Target Interval = [66, 72] → [μ - 2σ, μ + 1σ]

Resulting Percentage

81.5% (13.5% + 34% + 34%)

Time (min):

Standard Normal (Z) Distribution

Key Properties & Definition

A special normal distribution characterized by:

• Mean μ = 0
• Standard deviation σ = 1

Used to compare values from different distributions by converting raw data into standardized z-scores.

Z-Score Formula

Standardize any normal distribution value:

Interpretation & Worked Example

Interpretation Rules: Z = 0 (at mean) | Z > 0 (above mean) | Z < 0 (below mean)

Example Calculation:

Given: Mean μ = 100, Std Dev σ = 15, Score X = 115

Z = (115 − 100) / 15 = 1.0

Meaning: Score is 1 standard deviation above mean, indicating better-than-average performance.

For a sufficiently large sample size n, the distribution of the sample mean is approximately normal, regardless of the population’s original distribution.

SUMMARY

What Should You Leave With?

CMSC 320 • Basic Probability & Distributions

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Conditioning

New information changes the relevant probability.

Bayes

Update beliefs using evidence.

Distributions

Match the model to the data-generating process.

Expected value

Compare long-run outcomes and decisions.

Normal + Z

Reason about center, spread, and standardized distance.

CLT

Explains why sample means support statistical inference.

Next step: use these ideas to describe and reason about real datasets.