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Lesson 39 — Basic Discrete Probability

Sample space, events, Kolmogorov axioms. Classical probability: favorable cases over possible. Conditional probability, Bayes.

Used in: 1.º ano do EM (15–16 anos) · Equiv. Math B japonês · Equiv. Stochastik Klasse 11 alemã · Equiv. H2 Math Statistics (Singapura)

P(A) = \frac{|A|}{|\Omega|}, \qquad P(A|B) = \frac{P(A \cap B)}{P(B)}

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Rigorous notation, full derivation, hypotheses

Axioms and formulas

Sample space

$\Omega$ : set of all possible outcomes of the experiment. Event: subset of $\Omega$ .

Kolmogorov axioms (1933)

$P(A) \geq 0$ for every event $A$ .
$P(\Omega) = 1$ .
$\sigma$ -additivity: if $A_1, A_2, \ldots$ disjoint: $P(\bigcup A_i) = \sum P(A_i)$ .

Classical probability (equiprobable space)

$P(A) = \frac{|A|}{|\Omega|}$

Immediate properties

$P(\emptyset) = 0$ .
$P(A^c) = 1 - P(A)$ .
$P(A \cup B) = P(A) + P(B) - P(A \cap B)$ (inclusion-exclusion).
$A \subseteq B \Rightarrow P(A) \leq P(B)$ .
$P(A) \in [0, 1]$ .

Conditional probability

$P(A | B) = \frac{P(A \cap B)}{P(B)}, \quad P(B) > 0$

Independence

$A$ and $B$ are independent if $P(A \cap B) = P(A) P(B)$ (equivalent: $P(A|B) = P(A)$ ).

Bayes' theorem

$P(A | B) = \frac{P(B | A) P(A)}{P(B)}$

Total probability

If $\{A_i\}$ partition of $\Omega$ : $P(B) = \sum_i P(B | A_i) P(A_i)$

Discrete random variable

$X: \Omega \to \mathbb R$ . Distribution: $P(X = x_i) = p_i$ with $\sum p_i = 1$ .

Distribution	Formula	Appearance
Bernoulli	$P(X=1) = p$	1 trial
Binomial	$\binom{n}{k}p^k(1-p)^{n-k}$	$n$ trials
Geometric	$(1-p)^{k-1}p$	waiting time
Poisson	$e^{-\lambda}\lambda^k/k!$	rare events

Expectation and variance

$E[X] = \sum x_i p_i$
$\text{Var}(X) = E[(X - E[X])^2] = E[X^2] - E[X]^2$

Exercise list

46 exercises · 11 with worked solution (25%)

Application 34Understanding 2Modeling 8Challenge 1Proof 1

Ex. 39.1Application
Toss a coin. $P(\text{heads})$ . (Ans: $1/2$ .)
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Ex. 39.2Application
Roll a die. $P(\text{even})$ . (Ans: $1/2$ .)
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Ex. 39.3Application
Roll a die. $P(X > 4)$ . (Ans: $1/3$ .)
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Ex. 39.4Application
Toss 2 coins. $P(\text{both heads})$ . (Ans: $1/4$ .)
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Ex. 39.5Application
Roll 2 dice. $P(\text{sum 7})$ . (Ans: $6/36 = 1/6$ .)
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Ex. 39.6Application
Roll 2 dice. $P(\text{sum } > 9)$ . (Ans: $6/36 = 1/6$ .)
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Ex. 39.7Application
Draw 1 card from a deck. $P(\text{king})$ . (Ans: $4/52 = 1/13$ .)
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Ex. 39.8Application
Draw 1 card. $P(\text{hearts})$ . (Ans: $1/4$ .)
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Ex. 39.9Application
Draw 2 cards without replacement. $P(\text{both kings})$ . (Ans: $\binom{4}{2}/\binom{52}{2} = 1/221$ .)
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Ex. 39.10Application
Draw 1 card. $P(\text{king OR hearts})$ . (Ans: $16/52 = 4/13$ .)
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Ex. 39.11Application
$P(A) = 0.3$ , $P(B) = 0.5$ , $P(A \cap B) = 0.1$ . $P(A \cup B)$ ? (Ans: $0.7$ .)
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Ex. 39.12Application
$P(A^c)$ if $P(A) = 0.7$ . (Ans: $0.3$ .)
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Ex. 39.13Application
Show $P(A \cap B) \leq \min(P(A), P(B))$ .
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Ex. 39.14Application
$P(A) = 0.6$ , $P(B|A) = 0.4$ . $P(A \cap B)$ ? (Ans: $0.24$ .)
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Ex. 39.15ApplicationAnswer key
$P(A) = 0.5$ , $P(B) = 0.3$ , $A, B$ independent. $P(A \cap B)$ ? (Ans: $0.15$ .)
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Ex. 39.16Application
Roll a die twice. $P(\text{first 6, second any})$ . (Ans: $1/6$ .)
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Ex. 39.17ApplicationAnswer key
Mega-Sena: $P(\text{hit 6})$ . (Ans: $1/\binom{60}{6} \approx 2 \times 10^{-8}$ .)
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Ex. 39.18Application
Quina: $P(\text{hit 5 of 6 marked})$ .
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Ex. 39.19Application
Binomial: $X \sim \text{Bin}(10, 0.5)$ . $P(X = 5)$ ? (Ans: $\binom{10}{5}/2^{10} \approx 0.246$ .)
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Ex. 39.20ApplicationAnswer key
$X \sim \text{Bin}(5, 0.3)$ . $P(X = 2)$ ? (Ans: $\binom{5}{2} \cdot 0.3^2 \cdot 0.7^3 \approx 0.309$ .)
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Ex. 39.21ApplicationAnswer key
$P(A) = 0.4, P(B) = 0.5, P(A \cap B) = 0.2$ . $P(A|B)$ ? (Ans: $0.4$ .)
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Ex. 39.22ApplicationAnswer key
In the exercise above, are $A$ and $B$ independent? (Ans: yes, $P(A|B) = P(A)$ .)
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Ex. 39.23Application
2 dice. $P(\text{sum 7} | \text{first is 4})$ . (Ans: $1/6$ .)
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Ex. 39.24ApplicationAnswer key
Box with 3 white and 7 black. Draw 2 without replacement. $P(\text{both white})$ . (Ans: $\binom{3}{2}/\binom{10}{2} = 1/15$ .)
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Ex. 39.25Application
$P(\text{both black})$ in the same problem. (Ans: $\binom{7}{2}/\binom{10}{2} = 21/45 = 7/15$ .)
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Ex. 39.26Application
$P(\text{1 white, 1 black})$ . (Ans: $\binom{3}{1}\binom{7}{1}/\binom{10}{2} = 7/15$ .)
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Ex. 39.27ApplicationAnswer key
3 coins. $P(\text{exactly 2 heads})$ . (Ans: $3/8$ .)
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Ex. 39.28Application
$X \sim \text{Bin}(20, 0.1)$ . $P(X \geq 1)$ . (Ans: $1 - 0.9^{20}$ .)
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Ex. 39.29Application
Apply Bayes: $P(A) = 0.4, P(B|A) = 0.8, P(B|A^c) = 0.3$ . $P(A|B)$ ? (Ans: $0.32/0.5 = 0.64$ .)
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Ex. 39.30ApplicationAnswer key
Total probability: $P(A_1) = 0.3, P(A_2) = 0.5, P(A_3) = 0.2$ , $P(B|A_i) = 0.9, 0.5, 0.1$ . $P(B)$ ? (Ans: $0.57$ .)
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Ex. 39.31Application
Roll 2 dice. $P(\text{any 6})$ . (Ans: $11/36$ .)
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Ex. 39.32Application
In a class, $60\%$ are girls, $40\%$ boys. It's known that $80\%$ of girls and $50\%$ of boys passed. A student passed: $P(\text{girl})$ ?
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Ex. 39.33Application
$X \sim \text{Bin}(8, 0.25)$ . $E[X]$ and $\text{Var}(X)$ . (Ans: $E = 2, V = 1.5$ .)
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Ex. 39.34Application
Expectation of rolling 1 die. (Ans: $3.5$ .)
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Ex. 39.35ModelingAnswer key
A/B testing: $10\%$ see version B. Probability of 3 friends seeing B (independent)? (Ans: $0.1^3 = 0.001$ .)
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Ex. 39.36Modeling
Rare disease: $P(D) = 0.01$ . Test: sensitivity $95\%$ , specificity $90\%$ . $P(D|+)$ ?
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Ex. 39.37ModelingAnswer key
Quality control, defect rate $2\%$ . $P(\text{0 defects in 50 samples})$ . (Ans: $0.98^{50} \approx 0.364$ .)
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Ex. 39.38Modeling
In a spam filter, $P(\text{spam}|\text{contains viagra}) > 0.9$ via Bayes — model.
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Ex. 39.39Modeling
In a card game, probability of two pairs (5 cards). Compute via combinatorics.
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Ex. 39.40Modeling
Birthday: 23 people, $P(\text{2 share a birthday}) > 0.5$ . Compute explicitly.
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Ex. 39.41Modeling
In a computer network, probability of end-to-end connection in series of 5 links, each with $99\%$ reliability.
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Ex. 39.42ModelingAnswer key
In an ML classifier, false positive $5\%$ , false negative $2\%$ , prevalence $1\%$ . $P(\text{true positive}|\text{positive test})$ .
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Ex. 39.43Understanding
Show $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ via Kolmogorov.
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Ex. 39.44Understanding
Show that if $A$ and $B$ are independent, then $A^c$ and $B^c$ are also independent.
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Ex. 39.45Challenge
Monty Hall: 3 doors, 1 has the prize. You pick one; the host opens one of the other two without prize. Do you switch? What's the probability of winning by switching? (Ans: $2/3$ .)
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Ex. 39.46Proof
Prove Bayes' theorem from the definition of conditional probability.

Sources

OpenIntro Statistics — Diez, Çetinkaya-Rundel, Barr · 2019, 4th ed · EN · CC-BY-SA · ch. 3: probability. Primary source.
Introduction to Probability — Blitzstein, Hwang · 2019, 2nd ed · EN · free (authors) · ch. 1-2: counting and Bayes.
Introductory Statistics — Illowsky, Dean (OpenStax) · 2022, 2nd ed · EN · CC-BY-NC-SA · ch. 3.