Formula Sheet — Probability

Card 1: Event vocabulary

Term Meaning
Event any subset E⊆SE \subseteq S
Impossible / sure event ϕ\phi (probability 0) / SS (probability 1)
Simple / compound event one sample point / more than one
'not AA' A′=S−AA' = S - A
'AA or BB' / 'AA and BB' A∪BA \cup B / A∩BA \cap B
'AA but not BB' A−B=A∩B′A - B = A \cap B'
Mutually exclusive A∩B=ϕA \cap B = \phi
Exhaustive E1∪⋯∪En=SE_1 \cup \cdots \cup E_n = S

Card 2: Axioms and computation

Rule Statement
Axioms P(E)≥0P(E) \geq 0; P(S)=1P(S) = 1; P(E∪F)=P(E)+P(F)P(E \cup F) = P(E) + P(F) for disjoint E,FE, F
Sample points 0≤P(ωi)≤10 \leq P(\omega_i) \leq 1, ∑P(ωi)=1\sum P(\omega_i) = 1, P(A)=∑ωi∈AP(ωi)P(A) = \sum_{\omega_i \in A} P(\omega_i)
Equally likely P(E)=n(E)n(S)P(E) = \frac{n(E)}{n(S)}
Addition rule P(A∪B)=P(A)+P(B)−P(A∩B)P(A \cup B) = P(A) + P(B) - P(A \cap B)
Complement P(A′)=1−P(A)P(A') = 1 - P(A); P(ϕ)=0P(\phi) = 0

Card 3: JEE Corner

Tool Statement
Atleast one P(atleast one)=1−P(none)P(\text{atleast one}) = 1 - P(\text{none})
De Morgan P(A′∩B′)=1−P(A∪B)P(A' \cap B') = 1 - P(A \cup B); P(A′∪B′)=1−P(A∩B)P(A' \cup B') = 1 - P(A \cap B)
Only AA P(A∩B′)=P(A)−P(A∩B)P(A \cap B') = P(A) - P(A \cap B)
Exactly one of A,BA, B P(A)+P(B)−2P(A∩B)P(A) + P(B) - 2P(A \cap B)
Three events add singles −- pairs ++ triple
Bounds max⁡(0,P(A)+P(B)−1)≤P(A∩B)≤min⁡(P(A),P(B))\max(0, P(A){+}P(B){-}1) \leq P(A \cap B) \leq \min(P(A), P(B))
Odds in favour a:ba : b P=aa+bP = \frac{a}{a+b} (against a:ba : b: P=ba+bP = \frac{b}{a+b})
Derangements D3=2D_3 = 2, D4=9D_4 = 9

Last-Minute Mistake Checklist

  1. Check equal likelihood before using n(E)n(S)\frac{n(E)}{n(S)} — dice sums and 'number of heads' are NOT equally likely; ordered pairs and toss-strings are.
  2. A valid assignment needs BOTH tests: every P(ωi)∈[0,1]P(\omega_i) \in [0, 1] AND total exactly 1 — a clean-looking list summing to 2.1 is invalid.
  3. Never add probabilities of overlapping events without subtracting P(A∩B)P(A \cap B) — plain addition is only for mutually exclusive events.
  4. 'Or' means union, 'and' means intersection — translate word problems into sets before computing.
  5. 'Atleast one' goes through the complement; forgetting costs minutes and invites case-counting errors.
  6. De Morgan directions: 'neither… nor' =(A∪B)′= (A \cup B)', 'not both' =(A∩B)′= (A \cap B)' — do not swap them.
  7. P(A∩B)≤min⁡(P(A),P(B))P(A \cap B) \leq \min(P(A), P(B)) always — use it to catch inconsistent data (and inconsistent answers).
  8. Combinations for hands and committees, permutations for orders — a 7-card hand is unordered; a relay finish is ordered.
  9. Repeated letters divide the count: arrangements of MATHEMATICS are 11!2!2!2!\frac{11!}{2!2!2!}, not 11!11!.
  10. Mutually exclusive is not the same as exhaustive — disjointness says events cannot overlap; exhaustiveness says they cover SS. Check each separately.

How this chapter flows on: Class 11 mathematics ends here. Class 12 probability starts from these axioms and adds conditional probability, independence, Bayes' theorem and random variables — the addition rule and complement tricks stay in daily use throughout.