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.

Week map

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.

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

Deep study

Statistics I · Week 7 — Probability axioms

Week map

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.

Axioms (finite setting)

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

General addition (not necessarily disjoint)

Pattern families

Easy — Equally likely counting

Medium — Disjoint union

Hard — Inclusion-exclusion light

Worked mini-examples

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)

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

Pattern families

Easy — Complement

Medium — Disjoint union

Hard — Counting probability

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

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

Practice loop

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

Related

• Full weeks 1–8 prep guide
• Pattern atlas note
• Formula chain desk