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# Statistics I · Week 7 — Probability axioms Deep study for Quiz 2 week 7. Probability assigns numbers to outcomes between 0 and 1, governed by addition for disjoint events and normalization over the sample space.

Statistics I · Week 7 — Probability axioms

Deep study for Quiz 2 week 7. Probability assigns numbers to outcomes between 0 and 1, governed by addition for disjoint events and normalization over the sample space.

Week map

Sample space → event → probability scale → axioms → equally likely outcomes → complement rule → union of disjoint events.

Probability notation

$S$ → sample space → set of all possible outcomes.
$E$ → event → subset of $S$ .
$P(E)$ → probability of event $E$ → number in $[0, 1]$ .
$P(E) = \dfrac{\text{favorable outcomes}}{\text{total outcomes}}$ when outcomes equally likely and finite.

Mini-example: fair die.

S = \{1,2,3,4,5,6\}

. Event “even”

= \{2,4,6\}

P(\text{even}) = 3/6 = 1/2

Axioms (finite setting)

$P(E) \geq 0$ for any event $E$ .
$P(S) = 1$ .
If $A$ and $B$ disjoint: $P(A \cup B) = P(A) + P(B)$ .

Complement:

P(A^c) = 1 - P(A)

where

A^c

is “not

A

.”

Mini-example:

P(\text{not even on die}) = 1 - 1/2 = 1/2

General addition (not necessarily disjoint)

P(A \cup B) = P(A) + P(B) - P(A \cap B)

Subtract overlap once.

Pattern families

Easy — Equally likely counting

List

S

, count favorable, divide. Use complement when “at least one” is easier.

Medium — Disjoint union

Add probabilities of mutually exclusive events. Verify they do not overlap.

Hard — Inclusion-exclusion light

Two events with overlap: add singles, subtract intersection. Translate word problem to

A

B

A \cap B

Worked mini-examples

Example 1 — Die.

P(\text{roll } 5) = 1/6

Example 2 — Complement.

Draw one card.

P(\text{not heart}) = 1 - 13/52 = 3/4

Example 3 — Disjoint sum.

Die:

A = \{1,2\}

B = \{5,6\}

disjoint.

P(A \cup B) = 2/6 + 2/6 = 4/6

Example 4 — Inclusion-exclusion.

P(A)=0.4

P(B)=0.3

P(A \cap B)=0.1

P(A \cup B)=0.4+0.3-0.1=0.6

Example 5 — Invalid probability check.

P(A)=0.7

and

P(B)=0.5

and they are disjoint,

P(A \cup B)=1.2

— impossible; they cannot be disjoint.

Traps

Probabilities outside $[0,1]$ or sum of all outcomes not 1.
Adding $P(A)+P(B)$ when events overlap without subtracting intersection.
Assuming equally likely when outcomes are not (biased coin).
Complement of wrong event — define $A^c$ clearly.
Confusing probability with odds.

Diagnostic (try yourself)

Fair coin flipped once. What is $P(\text{heads})$ ?
Two disjoint events with $P(A)=0.3$ , $P(B)=0.45$ . Find $P(A \cup B)$ .
$P(E) = 0.82$ . What is $P(E^c)$ ?
Bag: 3 red, 5 blue balls, one draw. $P(\text{red})$ ?
$P(A)=0.5$ , $P(B)=0.4$ , $P(A \cap B)=0.2$ . Find $P(A \cup B)$ .