Introduction to Probability — Detailed Notes
Probability measures the likelihood of an event, just as length measures distance. It deals with random situations — ones where all possible outcomes are known, but which one will occur cannot be predicted in advance (tossing a coin, rolling a die, a lucky draw).
7.1 The probability scale
0 ≤ P(E) ≤ 1 — 0 = impossible, ½ = even chance, 1 = certain
Words along the scale: impossible → less likely → even chance → more likely → certain. A probability of 0.75 means a 75% chance.
7.2 Measuring probability objectively
Experimental probability = (number of times the event occurred) ÷ (total number of trials) — based on actual data (the relative frequency).
Theoretical probability P(A) = (number of favourable outcomes) ÷ (number of possible outcomes) — assumes all outcomes are equally likely (a fair situation).
By the Law of Large Numbers, as the number of trials grows, the experimental probability gets closer to the theoretical one. Statistical probability from a sample is used to estimate values for a whole population (e.g. 40% of a sample like mango → ≈ 40% of the school). A fair/unbiased coin or die has no reason to favour any outcome.
Gambler’s Fallacy: each toss/roll is independent — the coin/die has “no memory.” After six heads, the next toss is still P(tails) = ½. Past results do not change future independent events.
7.3 Sample space & events
The sample space S lists every possible outcome once; its size is n(S). Examples: coin S = {H, T}; die S = {1,2,3,4,5,6}; two coins S = {HH, HT, TH, TT}. An event is a subset of the sample space (e.g. ‘at least one head’ = {HH, HT, TH}).
7.4 Tree diagrams & geometric probability
A tree diagram lists all outcomes of a multi-step experiment — each path from start to end is one complete outcome (e.g. tossing a coin twice gives 4 paths). For a region, geometric probability = (favourable area) ÷ (total area).
Key Terms
probability, random experiment, outcome, trial, probability scale, experimental probability, relative frequency, theoretical probability, equally likely, Law of Large Numbers, sample & population, fair/unbiased, Gambler’s Fallacy, sample space, event, tree diagram.
End-of-Chapter Exercises — Full Solutions
Complete worked solutions to all 16 end-of-chapter problems (starred * = HOTS), plus key in-text exercise answers.
1Fill in the blanks.
(i) The probability of an impossible event is 0. (ii) The set of all possible outcomes is the sample space. (iii) The probability of a certain event is 1. (iv) Tossing a fair coin has probability ½ for heads.
2Of 50 students, 15 like football. State the term and its value.
The number 15 is the frequency; the relative frequency = 15/50 = 3/10 = 0.3 (this is the experimental probability that a randomly chosen student likes football).
3Which experiments have equally likely outcomes? Explain.
(i) Car starts / does not start — not equally likely (depends on the car’s condition). (ii) Tossing a fair coin — equally likely (½ each). (iii) Rolling a fair die — equally likely (⅙ each). (iv) 3 red, 7 blue marbles — not equally likely (blue is more likely). (v) A baby is a boy or girl — taken as (approximately) equally likely.
4Write the sample space and the probability.
(i) Two coins, at least one head: S = {HH, HT, TH, TT}; favourable = 3 → 3/4. (ii) Cards 1–10, even: {2,4,6,8,10} = 5 → 5/10 = 1/2. (iii) Die > 4: {5,6} = 2 → 2/6 = 1/3. (iv) 3 red, 2 blue, 1 green (6 balls), not red: 3 → 3/6 = 1/2. (v) Three coins, exactly two heads: {HHT, HTH, THH} = 3 of 8 → 3/8.
5A bag has 3 candies: strawberry, lemon, mint. Find P(strawberry).
One favourable out of three equally likely: P = 1/3.
6List all outfits from 2 shirts (red, blue) and 3 pants (jeans, khakis, shorts) in a table.
6 combinations: Red–Jeans, Red–Khakis, Red–Shorts, Blue–Jeans, Blue–Khakis, Blue–Shorts (2 × 3 = 6 outfits).
7Tyre distances in 1000 cases: <4000 → 20, 4001–9000 → 210, 9001–14000 → 325, >14000 → 445. Find P(lasts …).
(i) Less than 4000 km: 20/1000 = 0.02. (ii) Between 4000 and 14000 km: (210 + 325)/1000 = 535/1000 = 0.535. (iii) More than 14000 km: 445/1000 = 0.445.
8Cards spell P, E, A, C, E. Find (i) P(P, E or C), (ii) P(not an E).
There are 5 cards. (i) P, E or C = P(1) + E(2) + C(1) = 4 → 4/5. (ii) Not an E = P, A, C = 3 → 3/5.
9(*) A spinner points equally likely at 1–8. Find P of pointing at:
(i) 8 → 1/8. (ii) odd {1,3,5,7} → 4/8 = 1/2. (iii) > 2 {3,4,5,6,7,8} → 6/8 = 3/4. (iv) < 9 (all) → 8/8 = 1 (certain). (v) multiple of 3 {3,6} → 2/8 = 1/4.
10(*) 4 red and 5 blue balls; one is drawn and set aside, then a second is drawn. Use a tree diagram.
First draw from 9 balls, second from the remaining 8. (i) P(red then blue) = (4/9)(5/8) = 20/72 = 5/18. (ii) P(two blue) = (5/9)(4/8) = 20/72 = 5/18.
11(*) Throwing a pair of dice: give an event with probability 0 and one with probability 1.
Probability 0 (impossible): ‘the sum is 13’ (the maximum sum is 12). Probability 1 (certain): ‘the sum is between 2 and 12’ — every throw satisfies this.
12(*) Sample space and probability.
(i) Two dice, sum a prime > 5 (i.e. 7 or 11): sum 7 (6 ways) + sum 11 (2 ways) = 8 of 36 → 2/9. (ii) 4 red, 3 green, 2 blue (9 balls), two drawn (no replacement) of different colours: total pairs C(9,2)=36; same-colour pairs = C(4,2)+C(3,2)+C(2,2)=6+3+1=10; different = 26 → 26/36 = 13/18. (iii) Three coins, first is heads AND exactly two heads: {HHT, HTH} = 2 of 8 → 1/4. (iv) 4-digit number from 1,2,3,4 (no repeat), even: total 4! = 24; even → last digit 2 or 4 (2 × 3! = 12) → 12/24 = 1/2. (v) 3-question MCQ (4 options), exactly 2 correct by guessing: C(3,2)(1/4)²(3/4) = 3 × (1/16)(3/4) = 9/64.
13(*) A box has 4 balls numbered 1–4. Give the sample space (tree) and its size for:
(i) With replacement — second draw again from 4: size = 4 × 4 = 16. (ii) Without replacement — second draw from remaining 3: size = 4 × 3 = 12. (iii) Sizes are 16 and 12 respectively.
14(*) Simultaneously toss a coin and draw a card numbered 1–6. List the sample space.
S = {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6} — size 12 (2 × 6).
15(*) Three coins tossed; number of heads recorded. Which list is the sample space?
The number of heads can be 0, 1, 2 or 3, so the sample space is (iv) {0, 1, 2, 3}. The others fail: (i) {1,2,3} omits 0; (ii) {0,1,2} omits 3; (iii) {0,1,2,3,4} includes 4, which is impossible with three coins.
16(*) A dye is dropped at random on a 3 m × 2 m rectangle. Find P(it lands inside a circle of diameter 1 m).
Geometric probability = (area of circle)/(area of rectangle). Circle radius = 0.5 m → area = π(0.5)² = 0.25π. Rectangle area = 3 × 2 = 6. P = 0.25π/6 = π/24 ≈ 0.131.
IXKey in-text exercise answers (Sets 7.2 & 7.4).
Set 7.2 Q1 — green = 8/30 = 4/15 ≈ 0.27; in 600 sweets, yellow ≈ (7/30)×600 = 140. Set 7.2 Q2 — Arts Club = 11/40; Sports in 800 ≈ (9/40)×800 = 180 students. Set 7.2 Q5 — even number on a die {2,4,6} = 3/6 = 1/2. Set 7.4 Q1 — Basket A (apple 1/3), Basket B (banana 1/2): P(apple & banana) = 1/3 × 1/2 = 1/6.