1. Chapter at a Glance
Before a cricket match, the captains toss a coin — each has an equal chance of winning it. Probability is the branch of mathematics that measures such chances with numbers from 0 to 1. Its systematic study began in the 17th century with the work of Blaise Pascal and Pierre de Fermat on games of chance. In Class 9 you met experimental probability; this chapter focuses on theoretical (classical) probability for equally likely outcomes, with coins, dice, cards and bags of balls — a reliable source of marks in the board exam.
- Experiments, outcomes, sample space and events.
- Theoretical probability P(E) = favourable outcomes / total outcomes.
- Sure and impossible events; 0 ≤ P(E) ≤ 1.
- Complementary events: P(E) + P(not E) = 1.
- Elementary events; sum of their probabilities = 1.
- Problems with coins, dice (one and two), cards, bags and numbers.
2. Theoretical Probability
For an experiment whose outcomes are equally likely:
| Term | Meaning | Example |
|---|---|---|
| Sure event | Always happens; P = 1 | Getting a number < 7 on a die |
| Impossible event | Never happens; P = 0 | Getting 8 on a die |
| Elementary event | Event with only one outcome | Getting a 3 |
| Complement of E | "not E" | Not getting a 3 |
- 0 ≤ P(E) ≤ 1.
- P(E) + P(Ē) = 1, so P(Ē) = 1 − P(E).
- The probabilities of all elementary events of an experiment add up to 1.
3. Coins
| Coins | Sample space | Total |
|---|---|---|
| 1 | H, T | 2 |
| 2 | HH, HT, TH, TT | 4 |
| 3 | HHH, HHT, HTH, THH, HTT, THT, TTH, TTT | 8 |
Two coins are tossed. Find P(at least one head).
Solution: Favourable: HH, HT, TH = 3. P = 3/4. (Or 1 − P(TT) = 1 − 1/4.)
4. Dice
One die: 6 outcomes. Two dice: 6 × 6 = 36 outcomes (ordered pairs).
| Sum | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Ways | 1 | 2 | 3 | 4 | 5 | 6 | 5 | 4 | 3 | 2 | 1 |
Two dice are thrown. Find P(sum = 8), P(sum ≤ 3) and P(doublet).
Solution: Sum 8: 5/36. Sum ≤ 3: (1+2)/36 = 1/12. Doublet: 6/36 = 1/6.
5. Playing Cards
| Feature | Count |
|---|---|
| Total cards | 52 |
| Suits | 4: spades ♠ and clubs ♣ (black), hearts ♥ and diamonds ♦ (red), 13 each |
| Red / black | 26 / 26 |
| Face cards | 12 (J, Q, K of each suit) |
| Aces | 4 |
| Each rank (e.g. kings) | 4 |
One card is drawn from a well-shuffled deck. Find P(red king), P(face card), P(neither a heart nor an ace).
Solution: Red kings 2 → 1/26. Face cards 12 → 3/13. Hearts or aces = 13 + 3 = 16 → neither = 36 → 9/13.
6. Bags, Numbers and Spinners
A bag has 3 red, 5 black and 4 white balls. One is drawn at random. Find P(red), P(not black).
Solution: Total 12. P(red) = 1/4. P(black) = 5/12 → P(not black) = 7/12.
A number is chosen from 1 to 50. Find P(it is a perfect square).
Solution: 1, 4, 9, 16, 25, 36, 49 → 7/50.
7. Solved Problems (Board Pattern)
12 defective pens are accidentally mixed with 132 good ones. One pen is taken out at random. Find P(good pen).
Solution: 132/144 = 11/12.
A box has 5 red, 8 white and 4 green marbles. Find P(not green). How many blue marbles must be added so that P(blue) = 1/2?
Solution: P(not green) = 13/17. x/(17 + x) = 1/2 → 2x = 17 + x → x = 17.
A game: a coin is tossed 3 times. Hanif wins if all tosses show the same result; otherwise he loses. Find P(Hanif loses).
Solution: P(win) = 2/8 = 1/4 → P(lose) = 3/4.
8. Master Formula Table
| Rule | Formula |
|---|---|
| Classical probability | P(E) = n(E) / n(S) |
| Range | 0 ≤ P(E) ≤ 1 |
| Complement | P(Ē) = 1 − P(E) |
| Sure / impossible | 1 / 0 |
| n coins | 2ⁿ outcomes |
| Two dice | 36 outcomes |
| Deck | 52 cards; 12 face cards; 4 aces |
9. Master Summary Table
| Experiment | Total outcomes | Common questions |
|---|---|---|
| 1 coin / 2 coins / 3 coins | 2 / 4 / 8 | At least one head; exactly two heads |
| 1 die | 6 | Prime, even, > 4 |
| 2 dice | 36 | Sum, doublet, product |
| Cards | 52 | Face, red, king, ace, suit |
| Bag / numbers | Count total | Colour, multiples, squares |
Chart — P(sum) with two dice:
10. Mind Map
- Favourable / total
- Equally likely
- 0 to 1
- P(Ē) = 1 − P(E)
- Coins, dice
- Cards, bags
- List outcomes
- 6 × 6 grid
11. Tips and Tricks
- Write the sample space before counting favourable cases.
- Use the complement for "at least one" and "not" questions.
- For two dice, draw the 6 × 6 grid and shade the favourable squares.
- Counting (2, 3) and (3, 2) as the same outcome with two dice.
- Forgetting that face cards are only J, Q, K (not aces).
- Counting the "ace of hearts" twice in "heart or ace" questions.
12. Important One-liners
- The probability of an event lies between 0 and 1 inclusive.
- P(sure event) = 1; P(impossible event) = 0.
- P(E) + P(not E) = 1.
- Two dice have 36 equally likely outcomes.
- A standard deck has 12 face cards.
13. Exam-pattern Questions with Detailed Answers
Which of these cannot be the probability of an event?
- 2/3
- −1.5
- 15%
- 0.7
Show answer
Probability cannot be negative.
Answer: (B)
If P(E) = 0.05, what is P(not E)?
Show answer
1 − 0.05 = 0.95.
Answer: 0.95
A die is thrown once. Find P(a number less than 3).
Show answer
{1, 2}: 2/6 = 1/3.
Answer: 1/3
A card is drawn from a deck. Find P(a black face card).
Show answer
6 black face cards: 6/52 = 3/26.
Answer: 3/26
Two dice are thrown. Find P(the product of the numbers is 6).
Show answer
(1,6), (6,1), (2,3), (3,2): 4/36 = 1/9.
Answer: 1/9
A piggy bank has 100 coins of 50p, 50 coins of ₹1, 20 of ₹2 and 10 of ₹5. One coin falls out. Find P(it is a 50p coin) and P(it is not a ₹5 coin).
Show answer
Total 180. P(50p) = 100/180 = 5/9. P(not ₹5) = 170/180 = 17/18.
Answer: 5/9 and 17/18
A bag contains lemon-flavoured candies only. Find P(taking an orange candy) and P(taking a lemon candy).
Show answer
Orange: impossible → 0. Lemon: sure → 1.
Answer: 0 and 1
A number x is chosen from −2, −1, 0, 1, 2. Find P(x² < 2).
Show answer
x² < 2 for −1, 0, 1 → 3/5.
Answer: 3/5
A box has cards numbered 2 to 101. One card is drawn. Find P(an even number), P(a number less than 14) and P(a perfect square).
Show answer
Total 100. Even: 2, 4, …, 100 = 50 → 1/2. Less than 14: 2 to 13 = 12 → 3/25. Squares: 4, 9, …, 100 = 9 → 9/100.
Answer: 1/2, 3/25, 9/100
A bag has some red and blue balls; P(blue) = 2/5. If 6 more blue balls are added, P(blue) becomes 4/7. How many balls were in the bag originally?
Show answer
Let total = n, blue = 2n/5. (2n/5 + 6)/(n + 6) = 4/7 → 14n/5 + 42 = 4n + 24 → 18 = 6n/5 → n = 15.
Answer: 15 balls
Two dice are thrown. Find P(the difference of the numbers is 2).
Show answer
(1,3), (3,1), (2,4), (4,2), (3,5), (5,3), (4,6), (6,4) = 8 → 8/36 = 2/9.
Answer: 2/9
Two customers, Shyam and Ekta, visit a shop on the same week (Tuesday to Saturday). Each is equally likely to visit on any one of these days. Find P(they visit on the same day) and P(on consecutive days).
Show answer
Total 5 × 5 = 25. Same day: 5 → 1/5. Consecutive: (Tu,W), (W,Th), (Th,F), (F,Sa) and reverses = 8 → 8/25.
Answer: 1/5 and 8/25
14. Quick Revision Summary
- P(E) = favourable / total, for equally likely outcomes.
- 0 ≤ P(E) ≤ 1; sure = 1, impossible = 0.
- P(not E) = 1 − P(E).
- Coins 2ⁿ; two dice 36; deck 52 (12 face cards, 4 aces, 26 red).
- List the sample space carefully; ordered pairs for dice.