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Statistics — syllabus topic (vi) of 10 · 🎯 PYQs from this topic: 2024 Q76 - 2022 Q34

Part of the full combined notes. · 2 PYQ callouts inside.

🔷 TOPIC (vi) — THEORY OF PROBABILITY

Syllabus topic (vi) of 10 · runs until Topic (vii)

Covers: basic terms · types of events · the three definitions of probability · addition theorem · multiplication theorem · conditional probability · odds · Bayes theorem 🎯 Asked in the papers: 2024 · Q76 — which probability statements are correct · Panchayat 2020 · Q34 — P(Maths or Physics)

➕ Added — not in the original notes.

1. Basic Terms

Term	Meaning
Random experiment	An act whose outcome cannot be predicted with certainty (tossing a coin, rolling a die)
Trial	One performance of the experiment
Outcome	A single possible result
Sample space (S)	The set of ALL possible outcomes. Die: $S = \{1,2,3,4,5,6\}$ , so $n(S)=6$
Event (A)	Any subset of the sample space
Favourable outcomes	Outcomes that make the event happen

2. Types of Events

🎯 THIS CAME IN THE EXAM — 2024 · Q76

"Which statements about probability are correct?" → Answer: A (P and S) ⭐ This question is decided entirely by the definitions in the table below. The two traps it used:

"Mutually exclusive events always sum to 1" → ❌ FALSE — only if they are ALSO EXHAUSTIVE
"Probability can never be zero" → ❌ FALSE — an impossible event is exactly 0 ⚠️ Mutually exclusive · exhaustive · independent are three different things. The paper mixes them on purpose.

Type	Meaning	Example
Simple / Elementary	A single outcome	Getting a 4 on a die
Compound	More than one outcome	Getting an even number
Sure / Certain	Always happens → ⭐ P = 1	A number less than 7 on a die
Impossible	Can never happen → ⭐ P = 0	Getting 8 on a die
⭐ Mutually exclusive	Cannot happen together → $P(A \cap B) = 0$	Head and Tail on one toss
⭐ Exhaustive	Together cover the whole sample space → total probability = 1	{even, odd} on a die
⭐ Independent	One does not affect the other	Two separate coin tosses
Dependent	One does affect the other	Drawing 2 cards without replacement
Complementary ( $\bar{A}$ )	"A does not happen" → ⭐ $P(A) + P(\bar{A}) = 1$	Not getting a six
Equally likely	All outcomes have the same chance	A fair die

3. Definitions of Probability

(1) Classical / Mathematical (a priori): $P(A) = \frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}} = \frac{m}{n}$ Requires outcomes to be equally likely, mutually exclusive and exhaustive.

(2) Empirical / Statistical (a posteriori): based on actual repeated trials — $P(A) = \lim_{n \to \infty} \frac{\text{Number of times A occurred}}{n}$

(3) Axiomatic (Kolmogorov): ⭐ $0 \le P(A) \le 1$ · $P(S) = 1$ · for mutually exclusive events $P(A \cup B) = P(A) + P(B)$

⭐ THE RANGE — the most useful single fact

$\mathbf{0 \le P(A) \le 1}$ Impossible event = 0 · Certain event = 1 · Probability can NEVER be negative and NEVER exceed 1. ⭐ In the exam, any option greater than 1 (or negative) is instantly wrong.

4. Addition Theorem — "OR" / union

🎯 THIS CAME IN THE EXAM — Panchayat 2020 · Q34

"80 students: 30 opted Maths, 20 opted Physics, 10 opted both. Find P(Maths or Physics)." a) 1/2 ✅ · b) 1½ · c) 2½ · d) 3½ ⭐ Working: P(M) = 30/80, P(P) = 20/80, P(M∩P) = 10/80 P(M ∪ P) = 30/80 + 20/80 − 10/80 = 40/80 = 1/2 ⚠️ ⭐ Look at options b, c and d — every one of them is GREATER THAN 1, so none can be a probability. The range rule alone eliminates three of the four options before you calculate anything.

General (works always): $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

⭐ If A and B are mutually exclusive, $P(A \cap B) = 0$ , so: $P(A \cup B) = P(A) + P(B)$

Three events: $P(A \cup B \cup C) = P(A)+P(B)+P(C) - P(A \cap B) - P(B \cap C) - P(A \cap C) + P(A \cap B \cap C)$

5. Multiplication Theorem — "AND" / intersection

⭐ If A and B are INDEPENDENT: $P(A \cap B) = P(A) \times P(B)$

If DEPENDENT: $P(A \cap B) = P(A) \times P(B \mid A)$

6. Conditional Probability

$P(A \mid B) = \frac{P(A \cap B)}{P(B)}, \qquad P(B) \ne 0$ ⭐ If A and B are independent, $P(A \mid B) = P(A)$ — knowing B tells you nothing about A.

7. Odds

Odds in favour of A = $m : (n-m)$ = favourable : unfavourable
Odds against A = $(n-m) : m$
If odds in favour are $a : b$ then $P(A) = \dfrac{a}{a+b}$

8. Bayes Theorem

$P(A_i \mid B) = \frac{P(A_i) \cdot P(B \mid A_i)}{\sum P(A_j) \cdot P(B \mid A_j)}$ Used to revise a prior probability after new evidence arrives.

9. Worked Examples

Q1. (the Panchayat 2020 · Q34 type) In a class, P(passing Maths) = 2/5, P(passing Physics) = 3/10, P(passing both) = 1/5. Find P(passing Maths or Physics).

Sol Use the general addition theorem — $P(M \cup P) = P(M) + P(P) - P(M \cap P) = \frac{2}{5} + \frac{3}{10} - \frac{1}{5}$ $= \frac{4}{10} + \frac{3}{10} - \frac{2}{10} = \frac{5}{10} = \mathbf{\frac{1}{2}}$

Q2. A die is thrown once. Find the probability of getting an even number or a number greater than 4.

Sol $A = \{2,4,6\} \Rightarrow P(A) = \frac{3}{6}$ ; $B = \{5,6\} \Rightarrow P(B) = \frac{2}{6}$ ; $A \cap B = \{6\} \Rightarrow P(A \cap B) = \frac{1}{6}$ $P(A \cup B) = \frac{3}{6} + \frac{2}{6} - \frac{1}{6} = \frac{4}{6} = \mathbf{\frac{2}{3}}$

Q3. Two coins are tossed. Find P(at least one head).

Sol $S = \{HH, HT, TH, TT\}$ , $n(S) = 4$ . Easier by complement — $P(\text{at least one head}) = 1 - P(\text{no head}) = 1 - \frac{1}{4} = \mathbf{\frac{3}{4}}$ ⭐ "At least one" is almost always fastest via the complement.

Q4. A bag has 5 red and 3 black balls. Two are drawn with replacement. P(both red)?

Sol With replacement → independent — $P = \frac{5}{8} \times \frac{5}{8} = \mathbf{\frac{25}{64}}$ Without replacement (dependent): $\frac{5}{8} \times \frac{4}{7} = \frac{20}{56} = \frac{5}{14}$

🪤 EXAM TRAPS — these are what the statement-type questions test

Statement	Verdict
"Probability can never be zero"	❌ FALSE — an impossible event is exactly 0
"Probability of a certain event is 1"	✅ TRUE
"Mutually exclusive events always sum to 1"	❌ FALSE — ⭐ only if they are ALSO EXHAUSTIVE
"Mutually exclusive means independent"	❌ FALSE — opposites in effect: if A happens B cannot, so they are strongly dependent
"P(A) + P(not A) = 1"	✅ TRUE
"Probability can exceed 1 if there are many outcomes"	❌ FALSE — never
"For independent events P(A and B) = P(A) × P(B)"	✅ TRUE
"P(A or B) = P(A) + P(B) always"	❌ FALSE — only when mutually exclusive; otherwise subtract $P(A \cap B)$

🎯 IF YOU REMEMBER NOTHING ELSE FROM THIS TOPIC

⭐ $0 \le P \le 1$ — any option above 1 is eliminable free (this killed 3 of 4 options in Panchayat 2020 · Q34) ⭐ OR → add, then subtract the overlap · AND → multiply ⭐ Mutually exclusive ≠ exhaustive ≠ independent — the three words the paper mixes up on purpose

⬛ END OF TOPIC (vi) — Topic (vii) begins below.

🎯 PAA 2020 — PYQs on this topic

These came from the JKSSB Panchayat Accounts Assistant paper, 10 Nov 2020, and every one of them reappeared in the 2022 paper.

⭐ The official PAA answer key confirms this vault's answers on all 10 of its statistics/maths questions.

🎯 PAA 2020 · Q33 — Let us assume that astudent is choosen from a random Sample of 80 students out of which YOU have opted for Maths,20 opted for physics, and 10 have opted for both Maths and Physks. Find the probability that the student has opted for Either Maths or Physics?

a) ½ · b) 1 ½ · c) 2 ½ · d) 3 ½ → Ans: A ⭐ Reappeared as 2022·Q34. Use the addition theorem: P(A∪B) = P(A) + P(B) − P(A∩B).

🎯 FAA 2022 — PYQs on this topic

⭐ From the real FAA paper of 6 March 2022 — see FAA-2022-paper. Added 1 Sep 2026.

🎯 FAA 2022 · Q74 — "Ram and Shyam toss three unbiased coins. Shyam wins if he gets at least two tails. Find P(Shyam wins)."

a) 1/2 · b) 2/3 · c) 3/8 · d) 5/8 → Ans: A Sample space = 2³ = 8. "At least 2 tails" = exactly 2 (HTT, THT, TTH = 3) + exactly 3 (TTT = 1) = 4/8 = 1/2. 📌 The same paper's Mathematics section asked the mirror question (Q87: at most 2 heads = 1 − 1/8 = 7/8). ⭐ Probability is the one topic tested in BOTH the statistics and maths sections — worth double.