Module 7 Article: Quantifying Uncertainty — Bayes, Distributions, and the CLT

The big idea

Probability is the mathematics of uncertainty. In research it does two jobs:

1. It models the data-generating process (what outcomes are likely?).
2. It supports inference (how surprising is my result if nothing is going on?).

The two pillars you must internalise: Bayes' theorem (updating beliefs with evidence) and the Central Limit Theorem (sample means are normal). Everything else builds on these.

The probability calculus: from sets to numbers

Events, sample space, basic rules

• Sample space S: all possible outcomes.
• Event A ⊆ S: a subset of outcomes; P(A) = Σ P(outcome).
• Rules: 0 ≤ P(A) ≤ 1; P(S)=1; P(Aᶜ)=1−P(A).
• Addition: P(A∪B) = P(A)+P(B)−P(A∩B); mutually exclusive ⇒ no subtraction.

Worked example

Roll two dice. S = 36 outcomes.

• P(sum = 7) = 6/36 = 1/6.
• P(at least one 6) = 1 − P(no sixes) = 1 − (5/6)² = 11/36.

For our stress study: if 60% of students exercise, P(random student exercises) = 0.60.

Conditional probability and independence

• Conditional: P(A|B) = P(A∩B) / P(B).
• Independent: P(A|B) = P(A) ⇔ P(A∩B) = P(A)P(B).

Worked example

P(stressed) = 0.4; P(exercises | stressed) = 0.5; P(exercises | not stressed) = 0.7. P(stressed AND exercises) = 0.4 × 0.5 = 0.20.

Bayes' theorem

P(A|B) = P(B|A) · P(A) / P(B)
where P(B) = Σ_i P(B|A_i)·P(A_i)

"Posterior = likelihood × prior, normalised."

Worked example: medical screening

A stress-screening app has:

• prevalence P(D) = 0.01 (1% truly high-stress),
• sensitivity P(+|D) = 0.95,
• false positive P(+|¬D) = 0.10.

If a student screens positive, P(D|+) =

0.95 × 0.01 / (0.95×0.01 + 0.10×0.99)
= 0.0095 / (0.0095 + 0.099) = 0.0095/0.1085 ≈ 0.087

So a positive screen means only ~9% chance the student is truly high-stress — a dramatic illustration of the base-rate (prior) fallacy. Researchers must always ask: "How common is the condition in my population?"

Random variables

• Discrete RV: countable values (number of stressed students).
• Continuous RV: uncountable (stress score, time).
• Expected value: E[X] = Σ x·P(x); Variance: Var(X) = E[(X−μ)²].
• Linear properties: E[aX+bY] = aE[X]+bE[Y]; Var(aX) = a²Var(X).

Probability distributions

Discrete

Distribution	Models	Mean	Variance
Bernoulli(p)	one yes/no trial	p	p(1−p)
Binomial(n,p)	n independent yes/no	np	np(1−p)
Poisson(λ)	rare events in an interval	λ	λ

Continuous

Distribution	When	Mean	Variance
Uniform(a,b)	all values equally likely	(a+b)/2	(b−a)²/12
Normal(μ,σ²)	measurement noise, means	μ	σ²
t(df)	small-sample inference	0	df/(df−2)

Worked example: binomial stress count

If stress prevalence is 40%, in a sample of n = 50 the number stressed ~ Binomial(50, 0.4). Expected = 20, SD = √(50×0.4×0.6) ≈ 3.46.

P(exactly 20) = C(50,20)·0.4²⁰·0.6³⁰ ≈ 0.11. Use the normal approx (with continuity correction) for ranges like P(15 ≤ X ≤ 25).

The Central Limit Theorem (CLT)

CLT: For large n (typically n ≥ 30), the sampling distribution of the sample mean x̄ is approximately normal, regardless of the population's shape: x̄ ~ N(μ, σ²/n).

Why this drives the Z-test and t-test: even if individual stress scores are skewed, the mean of a sample of 30+ is approximately normal, so we can use normal-theory inference on the mean.

Demonstration (intuition)

Sample many groups of size n from a skewed population; plot the sample means. The histogram of means will look bell-shaped even though each individual value is skewed. That is the CLT in action.

Laws of large numbers and convergence

• Weak LLN: as n grows, x̄ converges in probability to μ.
• Strong LLN: x̄ converges almost surely to μ.
• Intuition: averages of repeated experiments settle on the true expectation.

Applications in research

Area	Probability tool
Diagnostic screening	Bayes (base-rate awareness)
Quality control	Binomial/Poisson counts
Significance testing	Sampling distribution of the mean (CLT) → Z/t
Sample-size/power	Binomial (power) and Normal (sample size)
Risk scoring	Conditional probability

Case study: probability in data analytics

A university's mental-health chatbot flags students whose weekly mood score drops below a threshold. The developers use the binomial distribution to set the threshold so each healthy student has <1% chance of being flagged (low false- positive rate), and Bayes' theorem to report, per flag, the posterior probability the student is truly at risk — combining the flag rate with the known prevalence of distress in the student body.

Common mistakes

• Confusing P(A|B) and P(B|A) (the prosecutor's / prosecutor's fallacy).
• Applying the normal approximation to a binomial with np < 5.
• Thinking "p = 0.03" means a 3% chance H₀ is true (it does not — see Module 8).
• Ignoring the prior (base rate) when interpreting a positive test.

Memory rules

• P(A∪B) = P(A)+P(B)−P(A∩B).
• Conditional P(A|B) = P(A∩B)/P(B).
• Bayes: posterior ∝ likelihood × prior; always mind the base rate.
• CLT: means are ~Normal for large n ⇒ Z/t inference on means.
• Binomial: np and n(1−p) should be ≥ 5 for the normal approximation.

Check your understanding

1. If P(A)=0.5 and P(B)=0.4 and A, B are independent, find P(A∩B).
2. What does the CLT guarantee about x̄?
3. Why should screening tests worry about the base rate?
4. For Binomial(n,p), when is the normal approximation safe?

Answers

Reveal answers after attempting the questions

1. P(A∩B) = P(A)P(B) = 0.5×0.4 = 0.20.
2. It is approximately normally distributed (regardless of the population shape), with mean μ and variance σ²/n.
3. Even a very sensitive/specific test can yield more false than true positives when the condition is rare.
4. When np ≥ 5 and n(1−p) ≥ 5.

See also

• Module 5 (sampling distribution of p̂/x̄).
• Module 8 (Z-test, t-test, p-values built on this).
• Glossary.md (probability, normal distribution, central limit theorem, Bayes' theorem).