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Statistics I · Week 7 — Probability

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Statistics I · Week 7 — Probability

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4 min read

2026-08-16

Reading compass

Now · Week map

Week 7 — probability

Quiz 2 scope: Weeks 1–8 per IITM May 2026 foundation courses. Source baseline: IITM BS admissions important-dates calendar · May 2026 cycle. Times on assessments are operational conventions — verify hall ticket.

Part of the Quiz 2 prep system · Weeks 1–8 index · Pattern atlas · Formula chains.

Week map

Sample space → event → rules → union/complement → disjoint add

Classify → Represent → Execute → Trap-check

Recognize: Ask: When can you add probabilities?
Procedure: Often easier P(at least one) = 1 − P(none). List none case carefully.
Variations / traps: Watch for: Adding P when events overlap.

Formula chain (compressed)

P(E) in [0,1] → complement → union for disjoint → general addition.

Probability — P(E) = favourable / total — equally likely outcomes
Complement — P(Eᶜ) = 1 − P(E) — at least one / not E
Union disjoint — P(A∪B)=P(A)+P(B) — mutually exclusive
General union — P(A)+P(B)−P(A∩B) — overlap possible
Sample space — all outcomes sum to 1 — sanity check

Open interactive formula desk · Week 7 tab.

Deep study

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 $P(A \cup B)$ .

ChatGPT prep archive

Chatty reference archive

Archived import for extra depth — complements the notes above, not official IITM material.

Core concepts

Sample space S: all outcomes; event: subset of S.
P(A): long-run proportion; 0 ≤ P ≤ 1; P(S)=1.
Complement: P(Aᶜ) = 1 − P(A).
Union: if disjoint, P(A∪B) = P(A)+P(B); else inclusion-exclusion.

Notation & vocabulary

Rule	Formula
complement	P(Aᶜ)=1−P(A)
disjoint union	P(A∪B)=P(A)+P(B)
general union	P(A)+P(B)−P(A∩B)

Pattern families

Easy — Complement

Often easier P(at least one) = 1 − P(none). List none case carefully.

Medium — Disjoint union

Verify events cannot both happen. Then add probabilities.

Hard — Counting probability

P(A) = (# favorable)/(# equally likely outcomes). Use perm/comb from week 6 for counts.

Drill these on the pattern atlas — filter to week 7.

Traps

Adding P when events overlap.
Probability > 1 from bad counting.
Assuming equally likely without justification.
Confusing AND with OR in wording.

Retrieval prompts

When can you add probabilities?
What is complement rule?
Define sample space.

Practice loop

Read Deep study (if present) or core concepts once.
Recite the formula chain without looking.
Open one easy pattern on the interactive atlas for week 7.
Attempt without solutions; mark studied after an honest try.
Say one trap aloud before closing the tab.