Introduction to Probability

Examples, Intuition, and Interactive Demos

Session roadmap

Probability language ──→ Bayes updating ──→ Test accuracy ──→ Simulation variability
    events + conditioning    priors + evidence   prevalence effects   simulation averages
           ╲                                      ╱
            ╲──── NBC: Bayes for classification ─╱

Five topics. One thread: how to reason consistently under uncertainty.

Spend 30 seconds here — this is the “where are we going” slide, not a content slide.

Point to each node: “Probability language = the vocabulary. Bayes = the updating engine. Sensitivity/specificity = a medical application. LLN = why simulations eventually settle. NBC connects Bayes to multi-feature classification and sits between the medical example and LLN.”

Students who read ahead will recognize all five terms; those who didn’t will get the orientation.

Pre-reading check

Hands up: true or false?

1. P(not A) = 1 − P(A)    true

2. “P(A | B) means: probability of B given A”    false

3. “A test with 99% sensitivity means a positive result is 99% likely correct”

   false

How to run this: Ask each question verbally, gather hands, THEN click to reveal the answer.

Q1 (true): Complement rule. Most students get this right. Quick validation.

Q2 (false): P(A | B) means probability of A given B. About 50% will get this wrong. The conditional direction (which event is “given”) is easy to confuse. Worth spending 30 seconds on.

Q3 (false): This is today’s key misconception (base-rate fallacy setup). Almost nobody gets this right on the first try. Don’t correct them yet — set up the suspense. “We’ll find the real answer in about 20 minutes.”

Q3 is the thread for the whole session. The misconception is the base-rate fallacy and is the central learning goal of the sensitivity/specificity section.

Warm-up: prior and posterior

Warm-up (5 min)

1. Which number is your starting belief?
2. Which numbers describe test quality?
3. Which number changes after a positive result?
4. Why can the updated probability still be low?

Open app in a new tab

Purpose: Activate prior/posterior vocabulary from the reading. Not a full exploration — 5 minutes only.

Visible slide intentionally stays minimal: just the four prompts. If students need the written follow-up, point them to Exercise 01 after class.

Common misconception: Students often say “the posterior is the test accuracy.” Redirect: “The posterior depends on prevalence too.”

Debrief question: “Which number did you change that moved the posterior the most?” Expected: Changing prevalence has the most dramatic effect on the posterior.

Transition: “You just saw that prevalence matters a lot. We’re going to understand why — using Bayes’ theorem.”

Probability language

What you must retain

• Probabilities are numbers between 0 and 1
• P(not A) = 1 − P(A) — complement rule
• P(A | B) means: probability of A given B
• Independence: knowing B does not change the probability of A

This is a reference slide — not new material.

Ask the class: “Give me one example of a conditional probability in your field or everyday life.” (Examples: probability of rain given cloudy sky; probability of test positive given the person has the disease.)

Foreshadow NBC: “The independence rule — knowing B changes nothing about A — is called the ‘naive’ assumption in Naive Bayes. We’ll see it again later.”

Jeffreys: axioms govern reasoning, not measurement

Jeffreys’ definition: probability is the degree of confidence that we may reasonably have in a proposition.

Frequentist definitions (von Mises, Fisher) claim probability is a frequency — measurement is built into the definition.

Jeffreys’ framework claims nothing about measurement — only about reasoning:

“Once you have accepted any probabilities, you must reason consistently with them.”

Bayes’ theorem follows as a consequence of consistent reasoning — not as an arbitrary formula.

Key contrast: Frequentists say probability is a frequency (you must run an experiment to have a probability). Jeffreys says probability quantifies reasonable confidence (no experiment required — but once you have numbers, consistency is mandatory).

Why this matters now: Students see Bayes’ theorem in a moment. The point of this slide is that Bayes is not a trick or a choice — it’s what consistent reasoning requires.

Advanced note (optional, if asked): Cox (1946) and Jaynes (2003) prove that any coherent measure of plausibility must satisfy the probability axioms. The axioms are unavoidable, not a convention.

Three rules you will use today

\[ \text{P}(\neg A) \;=\; 1 - \text{P}(A) \quad\quad \leftarrow \text{complement rule} \]

\[ \text{P}(A \mid B) \;=\; \frac{\text{P}(A \cap B)}{\text{P}(B)} \quad\quad \leftarrow \text{conditional probability ("given")} \]

\[ \text{P}(A \mid B) \;=\; \text{P}(A) \quad\quad \leftarrow \text{independence: B tells you nothing about A} \]

These three rules are sufficient for everything today.

Read each line aloud with the plain-English label.

Independence rule: “This third line — P(A|B) = P(A) — is the ‘naive’ assumption in Naive Bayes. We’ll use it in about 30 minutes.”

Students who read the handbook have seen this notation. This slide anchors the symbols they saw in print to their spoken equivalents. Spend about 60 seconds here.

Story: two sacks of coins

There are two sacks of coins.

• Sack 1: 150 gold, 50 silver   (75% gold)
• Sack 2: 100 gold, 200 silver   (33% gold)

A blindfolded person picks one sack at random and draws one coin. You observe a gold coin.

Question: which sack is now more plausible?

Set the scene verbally: “Imagine you’re in a room. There are two opaque bags in front of you…”

Do not reveal the handbook result yet — let students predict before computing.

Ask: “Before any calculation — which sack would you bet on? And roughly how confident are you, on a 0–1 scale?” Get a few verbal responses.

This grounds the subsequent app exploration: students have a verbal intuition to confirm or correct.

Reference result (for later reveal): \(\text{P}(\text{Sack 1} \mid \text{Gold}) = 9/13 \approx 0.692\).

Predict the direction first

Say this out loud before touching the app:

• “Gold is more common in Sack 1.”
• “So observing gold should increase my belief in Sack 1.”

This is a 60-second metacognitive pause. Get verbal responses from the class before moving to the app.

Typical correct response: “Sack 1, because gold is more common there.”

If a student gives the right direction for the right reason, validate it: “Exactly — and Bayes’ theorem will tell us how much to update, not just which direction.”

Then move to the app.

This is Bayes thinking in words: predict before computing.

Coin Sacks App

Task

1. Start with equal priors. Which sack is favored after seeing gold?
2. Verify: P(Sack 1 | Gold) ≈ 0.692.
3. Change the prior so Sack 2 is preferred 2:1. Does the conclusion change?
4. Make the two sacks nearly identical (same coin mix). What happens to the update?

Explain the update in words before reading numbers.

Open app in a new tab

What the app shows: Prior odds, likelihood ratio, posterior probability for each sack. Students set prior proportions and coin compositions.

Visible task card stays minimal. Story reminder: one drawn coin is gold. Point students to Exercise 03 after class for a written follow-up.

Common misconception: “The posterior is just the proportion of gold in Sack 1.” Redirect: “The posterior depends on the prior too. Change the prior and watch what happens.”

Debrief question: “What happens to P(Sack 1 | Gold) when both sacks have the same coin composition?” Expected: Posterior = Prior. Evidence with no differential likelihood carries zero information — the Bayes update is neutral.

Transition: “We saw this in the app. Now let’s name the pieces of that calculation.”

Bayes’ theorem: annotated

\[ \underbrace{\text{P}(A \mid B)}_{\text{posterior}} \;=\; \frac{ \underbrace{\text{P}(B \mid A)}_{\text{likelihood}} \;\times\; \underbrace{\text{P}(A)}_{\text{prior}} }{ \underbrace{\text{P}(B)}_{\text{evidence}} } \]

Read the formula aloud using plain-English labels: “posterior equals likelihood times prior, divided by evidence.”

Term labels (not on slide): posterior = updated belief, likelihood = compatibility of evidence with hypothesis, prior = starting belief, evidence = normalizer.

Coin example mapping: \(A\) = “Sack 1”, \(B\) = “Gold coin drawn”.

Emphasize the evidence term: “P(B) is just a scaling factor — it makes sure all the posterior probabilities add up to 1. In practice, you can often ignore it and just compare the numerators.”

Connect to the app: “In the app you just used, the slider for ‘prior’ is P(A), the coin compositions give the likelihoods, and the output is the posterior.”

Foreshadow sens/spec: “We’re about to apply this exact same formula to medical testing. The terminology changes — sensitivity, specificity — but the structure is identical.”

Story: screening for a rare condition

A financial fraud detection system has:

• Prevalence of fraud: 0.2% (2 in every 1,000 transactions)
• Sensitivity: 99% (correctly flags 99% of real fraud)
• Specificity: 99% (correctly clears 99% of legitimate transactions)

Question: if the system flags a transaction as fraudulent, how likely is it actually fraud?

Ask: “You have a 99% accurate test and it flags a transaction. What’s the probability of actual fraud?”

Prompt before the app: “Predict before computing.”

Most students will say “99%.” Write “99%?” on the board or note their prediction aloud.

This is the pre-reading check Q3 misconception. Students are about to discover why they got Q3 wrong — this moment should feel like a reveal.

Don’t give the answer yet. Say: “Let’s find out in the app.”

Positive Test App

Task

1. Set prevalence = 0.2%, sensitivity = 99%, specificity = 99%.
2. Record the result. Compare to your prediction from before.
3. Increase prevalence to 2%. What changes?
4. Reset. Increase specificity to 99.9%. What changes?
5. Write one sentence starting: “A flagged transaction means…”

Open app in a new tab

What the app shows: Prevalence, sensitivity, specificity sliders; PPV (Positive Predictive Value) as output.

Keep the visible task card focused on the 5 prompts. After the app work, direct students to Exercise 04 for a written follow-up. Reference result at baseline settings: PPV ≈ 16.6%.

Expected reaction: Students are surprised to see ~16.6%, not 99%. This surprise IS the learning. Let it land before explaining.

Common misconception: “99% accurate test means 99% chance the positive is real.” This is the base-rate fallacy. Name it after students have seen the result.

Debrief question: “If prevalence doubles to 0.4%, what happens to PPV?” Expected: PPV roughly doubles (from ~16.6% to ~28%).

Follow-up: “Which lever has the bigger effect — changing prevalence or changing specificity?” Answer: Prevalence dominates at very low base rates. Specificity becomes more important as prevalence rises.

What to say about a positive result

Use language like this:

• “The test is good, but fraud is rare.”
• “A flagged transaction increases the probability, but it may still be far from certain.”
• “To know how likely it is after a positive result, I need prevalence and test quality.”

Avoid saying:

• “The test is 99% accurate, so a flagged transaction is 99% likely fraud.”

The fraction of true positives among all positives (PPV) depends critically on prevalence.

This is a language slide. Help students find the correct words to describe what they just saw.

Practice verbally: Ask a student to read the first bullet. Then ask: “Can you say that in your own words without looking at the slide?”

Transition to NBC: “We just saw Bayes at work in a 1-feature problem: is the test positive or negative? Now we’ll apply the same logic to a classifier that uses multiple features — that’s Naive Bayes.”

Story: classifying a car’s origin

Cars93 dataset: 93 cars. Predict origin (USA / non-USA) from one feature: Man.trans.avail (Yes / No).

Man.trans.avail	P(feature | USA)	P(feature | non-USA)
No	0.542	0.133
Yes	0.458	0.867

Prior: P(USA) = 0.516. For a new car with Man.trans.avail = No:

• USA score \(\;\propto\;\) 0.516 × 0.542 = 0.280
• non-USA score \(\;\propto\;\) 0.484 × 0.133 = 0.064

Predicted: USA (P ≈ 81%).

The Cars93 dataset is built into R. Origin has two classes: USA (48 cars) and non-USA (45 cars). The feature Man.trans.avail (manual transmission available: Yes/No) is binary — a Bernoulli NBC.

Visible table values come from the app using the full dataset as training.

Intuition for the numbers: Most USA cars in this dataset do NOT offer manual transmission (0.542 No), while most non-USA cars DO (0.867 Yes). So observing “No manual available” is strong evidence for USA origin — likelihood ratio 0.542/0.133 ≈ 4.

Work through the arithmetic on the board. Ask: “Which likelihood ratio is more informative?” Answer: Man.trans.avail=No has ratio 4.1 (strongly USA); Man.trans.avail=Yes has ratio 0.458/0.867 ≈ 0.53 (modestly non-USA).

Key message: NBC applies to any binary classification — it is not tied to text/spam. Same formula, different domain.

State the assumption explicitly: NBC treats features as independent given the class (here there is only one feature, so the assumption is trivial).

Classifier App — Cars93

Task — select Origin + Man.trans.avail

1. Read the prior probabilities from the output. What do they represent?
2. Using the likelihood table, verify the prediction for a car with Man.trans.avail = No. Compute the scores by hand.
3. Set training set to 70%. Find sensitivity and specificity in the output. What do these numbers tell you about the classifier?
4. Use Shuffle Data several times. What changes: priors, sensitivity, specificity? Why? What does this variability remind you of from earlier today?

Open app in a new tab

Setup: Students select variables Origin and Man.trans.avail in the app. Origin is the dependent variable (class). Training set slider starts at 1 (100% training, no test set).

Task 1 answer: Prior = proportion of each class in the training data. P(USA) ≈ 0.516, P(non-USA) ≈ 0.484. Plain English: “About half the cars in the dataset are USA-made.”

Task 2 answer: USA score ∝ 0.516 × 0.542 = 0.280; non-USA score ∝ 0.484 × 0.133 = 0.064. Predicted: USA. P(USA | No) = 0.280/(0.280+0.064) ≈ 81%.

Task 3: With training = 70%, the remaining 30% (~28 cars) form the test set. Sensitivity = TP/(TP+FN) (proportion of USA cars correctly identified). Specificity = TN/(TN+FP) (proportion of non-USA correctly identified). Reference handbook table 8.1 (same confusion matrix as in the sensitivity/specificity section — same framework, now applied to a classifier). One-sentence training set explanation: “The training set estimates the model; the test set measures how well it performs on unseen data.”

Task 4: Each shuffle randomly assigns cars to training vs test. Prior probabilities, sensitivity, and specificity all change across shuffles because the training sample is a random subset of the 93 cars. This is the same variability as the hospital simulation — finite samples fluctuate. With more data (not possible here with 93 cars), these estimates would stabilize. That is the Law of Large Numbers applied to model evaluation.

After the live app work, direct students to Exercise 05 for a written follow-up.

Debrief question: “Why do sensitivity and specificity change when you shuffle but the likelihood table changes only slightly?” Expected: The likelihood table is estimated from the training set (which changes with each shuffle) — it does change, but the posterior probabilities are relatively robust because the likelihood ratios are large. Sensitivity/specificity are computed on the small test set, so they fluctuate more.

What is Naive Bayes?

One sentence:

Naive Bayes is Bayes’ theorem applied once per feature, assuming features are independent given the class.

Read the sentence aloud. Ask a student to repeat it without looking.

“Naive is not an insult to the method. It is a description of the assumption — the features are treated as independent.”

Visible slide intentionally stays one-sentence only. If asked, add: we will revisit interaction effects in the Gaussian NBC lecture.

This is a 30-second crystallization slide. Move on quickly.

Story: which hospital shows more extreme days?

Two hospitals record the proportion of boys born each day:

• Large hospital: ~45 births per day
• Small hospital: ~15 births per day

Question: which hospital more often records days with more than 60% boys?

\[ \bar{X}_n \;\xrightarrow{\;n \to \infty\;}\; \mu \quad\quad \text{(Law of Large Numbers)} \]

Predict before simulating.

This story is from Kahneman & Tversky (1972) — a classic demonstration of people ignoring sample size.

Ask: “Which hospital? Show of hands.” Expect: majority say large hospital, or a 50/50 split.

Correct answer: The small hospital, because with fewer births per day, the daily proportion fluctuates more. The expected proportion is the same (~0.51), but the variance around that expectation is larger for small samples.

Don’t reveal the answer yet. Send them to the simulation.

Hospital Simulation App

Task

1. Predict first: large or small hospital — which shows more extreme days?
2. Run the simulation. Was your prediction correct?
3. Increase the number of simulated days. What stabilizes?
4. Change the threshold (60% → 80%). What changes?
5. Explain using the words variability and sample size.

Open app in a new tab

What the app shows: Per-day birth proportions for a large and small hospital over simulated years; frequency of days exceeding a threshold.

Common misconception: “Both hospitals should show the same distribution — the true probability of a boy is the same.” Redirect: “Same expected value, very different variability. Small samples fluctuate more around the same mean.”

Debrief question: “If you run the simulation again, do you get identical numbers?” Expected: No — there is sampling variability across runs. “What would make the simulation more stable?” Expected: More simulated days. The Law of Large Numbers says the long-run average converges; it does not say each run is identical.

After the live app work, direct students to Exercise 06 for a written follow-up.

Key ideas

Topic	What to retain
Probability language	P(A|B) is conditional; independence means B tells you nothing about A
Jeffreys	Axioms govern reasoning, not measurement; Bayes follows from consistency
Bayes’ theorem	Posterior ∝ likelihood × prior; P(B) normalizes
Sensitivity / specificity	Test accuracy ≠ PPV; prevalence dominates at low base rates
Naive Bayes	Bayes per feature, independence assumed; “naive” is intentional
Law of Large Numbers	Sample means converge to μ; small samples fluctuate more

Go through each row. For each, ask the class to complete the “What to retain” cell from memory BEFORE reading the table.

If time permits (5+ minutes remaining): pairs. Each person explains one row to their partner in one sentence, then switch.

This is the consolidation slide — not new information.

Exit problem (pairs, 5 min)

A rapid test for a rare infection has:

• Prevalence: 0.5%
• Sensitivity: 98%
• Specificity: 95%

You test 1,000 people.

1. How many false positives do you expect?
2. What is the probability a positive result is a true infection (PPV)?
3. If you repeated this tomorrow with 1,000 new people, would the numbers be identical? Why not?

Pairs: 5 minutes. Walk around and listen to discussions.

Instruction (not on slide): solve first using handbook formulas, then verify with BayesExample2.

Expected answers: - True positives: 1000 × 0.005 × 0.98 ≈ 5 - False positives: 1000 × 0.995 × 0.05 ≈ 50 - Total positives: ~55, so PPV = 5/55 ≈ 9%

Q3 answer: No — random sampling means counts vary. The Law of Large Numbers says only the very long-run average stabilizes; individual runs of 1,000 people differ.

After 3 minutes, ask one pair for Q1 (expected ~50 false positives). Then reveal the PPV. End by pointing back to Q3 in the pre-reading check: “This was the answer to the question you got wrong at the start.”