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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 · 2022 · 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 — 2022 · 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 2022 · 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 2022 · 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.