←Back to the bookChapter 35 · Bayes' Theorem & the Law of Total Probability

Chapter 35 · Probability

Bayes' Theorem & the Law of Total Probability 🔄

Bayes' theorem reverses a conditional probability, turning P(evidence | cause) into P(cause | evidence). This notebook builds it from a tree, runs the Law of Total Probability that supplies its denominator, updates a prior into a posterior, accumulates evidence across two tests, and ends with a tiny naive Bayes classifier.

Statistics, Data Science and AI: A Visual Handbook · John Fisher · 2026

⚙️ Setup

import numpy as np
import matplotlib.pyplot as plt
rng = np.random.default_rng(34)
plt.rcParams.update({"figure.dpi":110,"font.size":11,"axes.spines.top":False,"axes.spines.right":False})
INDIGO="#4f46e5"; PINK="#db2777"; GREEN="#059669"
print("ready")

ready

DEMO 1 · BAYES FROM A TREE

Reverse the conditional

Two machines make the same part. Machine A makes 60% of parts at a 2% defect rate; Machine B makes 40% at a 5% defect rate. A part is defective. Which machine most likely made it?

# priors (which machine) and likelihoods (defect rate given machine)
pA, pB = 0.60, 0.40
p_def_A, p_def_B = 0.02, 0.05

# Law of Total Probability: the overall defect rate is a weighted average
p_def = pA*p_def_A + pB*p_def_B
print(f"P(defective) = 0.60*0.02 + 0.40*0.05 = {p_def:.3f}")

# Bayes: flip P(defect | machine) into P(machine | defect)
p_A_def = pA*p_def_A / p_def
p_B_def = pB*p_def_B / p_def
print(f"P(A | defective) = (0.60*0.02)/{p_def:.3f} = {p_A_def:.3f}")
print(f"P(B | defective) = (0.40*0.05)/{p_def:.3f} = {p_B_def:.3f}")
print(f"\nMost likely source of a defect: Machine {'B' if p_B_def>p_A_def else 'A'} ({max(p_A_def,p_B_def):.1%})")

P(defective) = 0.60*0.02 + 0.40*0.05 = 0.032
P(A | defective) = (0.60*0.02)/0.032 = 0.375
P(B | defective) = (0.40*0.05)/0.032 = 0.625

Most likely source of a defect: Machine B (62.5%)

Machine B makes fewer parts but is much more defect-prone, so among the defects it is the more likely source: P(B | defective) = 0.625. Notice the prior (40%) got updated by the evidence to 62.5%.

# verify by simulation
N=400_000
machine = rng.choice(["A","B"], size=N, p=[pA,pB])
defect  = rng.random(N) < np.where(machine=="A", p_def_A, p_def_B)
defects = machine[defect]
print(f"empirical P(B | defective) = {(defects=='B').mean():.3f}  (theory 0.625)")

empirical P(B | defective) = 0.622  (theory 0.625)

DEMO 2 · THE LAW OF TOTAL PROBABILITY

A weighted average over cases

The denominator of Bayes is the Law of Total Probability: split the world into cases that cover everything, then average the conditional probabilities, weighted by how likely each case is.

# P(defective) decomposed over the two machines
parts = {"from A": pA*p_def_A, "from B": pB*p_def_B}
total = sum(parts.values())
for k,v in parts.items():
    print(f"P(defective and {k}) = {v:.3f}")
print(f"P(defective) = sum = {total:.3f}")

fig,ax=plt.subplots(figsize=(7,1.7))
left=0
for (k,v),c in zip(parts.items(),[INDIGO,PINK]):
    ax.barh(0, v, left=left, color=c, edgecolor="white")
    ax.text(left+v/2, 0, f"{k}\n{v:.3f}", ha="center", va="center", color="white", fontweight="bold", fontsize=10)
    left+=v
ax.set_xlim(0,total); ax.set_yticks([]); ax.set_title("P(defective) = 0.032, split into the two machine contributions")
ax.set_xlabel("probability"); plt.tight_layout(); plt.show()

P(defective and from A) = 0.012
P(defective and from B) = 0.020
P(defective) = sum = 0.032

The Law of Total Probability says P(B) = Σ P(B | Aᵢ) P(Aᵢ) over a partition {Aᵢ}. Here the two machines partition every part, so the total defect rate 0.032 is just the sum of the two slices. That total is exactly the number Bayes divides by.

DEMO 3 · PRIOR → POSTERIOR

A medical test, the Bayesian way

A disease has 0.5% prevalence. A test has 99% sensitivity and 95% specificity. The prevalence is the PRIOR; the test result is the EVIDENCE; Bayes gives the POSTERIOR.

prior = 0.005                 # P(disease) before the test
sens  = 0.99                  # P(+ | disease)
spec  = 0.95                  # P(- | no disease)  ->  P(+ | no disease) = 0.05

p_pos = prior*sens + (1-prior)*(1-spec)        # Law of Total Probability
posterior = prior*sens / p_pos                 # Bayes
print(f"P(positive)            = {p_pos:.4f}")
print(f"prior     P(disease)   = {prior:.3f}")
print(f"posterior P(disease|+) = {posterior:.4f}  (about {posterior:.1%})")

P(positive)            = 0.0547
prior     P(disease)   = 0.005
posterior P(disease|+) = 0.0905  (about 9.0%)

A single positive test moves belief from 0.5% up to about 9%. That is an 18-fold jump, yet it is still far from certain, because the disease is rare. Bayes keeps the base rate in the picture where intuition tends to drop it.

DEMO 4 · EVIDENCE ACCUMULATES

Yesterday's posterior is today's prior

Bayesian updating is sequential: take a second, independent positive test. The posterior from the first test becomes the prior for the second.

def update(prior, sens, spec):
    p_pos = prior*sens + (1-prior)*(1-spec)
    return prior*sens / p_pos

belief = 0.005
print(f"start (prior)      P(disease) = {belief:.4f}")
for k in range(1,5):
    belief = update(belief, sens, spec)
    print(f"after {k} positive test(s): P(disease) = {belief:.4f}  ({belief:.1%})")

start (prior)      P(disease) = 0.0050
after 1 positive test(s): P(disease) = 0.0905  (9.0%)
after 2 positive test(s): P(disease) = 0.6633  (66.3%)
after 3 positive test(s): P(disease) = 0.9750  (97.5%)
after 4 positive test(s): P(disease) = 0.9987  (99.9%)

beliefs=[0.005]
for _ in range(5): beliefs.append(update(beliefs[-1], sens, spec))
fig,ax=plt.subplots(figsize=(7,3))
ax.plot(range(len(beliefs)), beliefs, "o-", color=INDIGO, lw=2)
ax.set_xlabel("number of independent positive tests"); ax.set_ylabel("P(disease)")
ax.set_title("Each positive test updates the posterior; evidence compounds")
for x,y in enumerate(beliefs): ax.annotate(f"{y:.0%}",(x,y),textcoords="offset points",xytext=(0,8),ha="center",fontsize=9)
ax.set_ylim(0,1.05); plt.tight_layout(); plt.show()

Two positive tests push P(disease) from 9% to about 66%; a third crosses 97%. No single result is conclusive, but independent evidence compounds, which is the whole engine of Bayesian learning.

DEMO 5 · BAYES IN MACHINE LEARNING

A tiny naive Bayes spam filter

A spam classifier is Bayes' theorem applied to words. Given the word 'free', is the email spam?

p_spam = 0.40                 # prior: 40% of mail is spam
p_free_spam = 0.80            # P("free" | spam)
p_free_ham  = 0.10            # P("free" | not spam)

p_free = p_spam*p_free_spam + (1-p_spam)*p_free_ham      # total probability
p_spam_free = p_spam*p_free_spam / p_free                # Bayes
print(f"P(\"free\")            = {p_free:.3f}")
print(f"prior     P(spam)      = {p_spam:.2f}")
print(f"posterior P(spam|free) = {p_spam_free:.3f}  (about {p_spam_free:.0%})")

P("free")            = 0.380
prior     P(spam)      = 0.40
posterior P(spam|free) = 0.842  (about 84%)

The word "free" lifts the spam probability from a 0.40 prior to about 0.84. A real naive Bayes filter multiplies many such word likelihoods together (the chain/multiplication rule from Chapter 31), each one nudging the posterior up or down. Bayes is the rule that turns features into a class probability.

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✅ What you built

You reversed a conditional with Bayes' theorem, computed its denominator with the Law of Total Probability, turned a prior into a posterior, accumulated evidence across repeated tests, and ran a one-word naive Bayes classifier. Every one of these is the same formula: posterior ∝ likelihood × prior.

Statistics, Data Science and AI: A Visual Handbook · © 2026 John Fisher

←Back to the bookChapter 35 · Bayes' Theorem & the Law of Total Probability