Probability
Why This Chapter Matters
Probability is the last chapter and has a guaranteed 3-5 marks in every board exam. The questions are logical rather than formula-heavy — they test whether you can identify sample spaces and favourable outcomes. With practice, this chapter becomes very scoring.
Prerequisites
Core Concepts
1. Theoretical (Classical) Probability
$$P(E) = \frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}$$
Important range: 0 ≤ P(E) ≤ 1
2. Complementary Events
If E is an event, then NOT E (written as Ē or E') is its complement.
$$P(E) + P(\bar{E}) = 1 \ \Rightarrow \ P(\bar{E}) = 1 - P(E)$$
3. Sample Space for Common Problems
#### Coin
Note: HT and TH are different (order matters)
#### Dice
Common two-dice events:
| Event | Favourable outcomes | Probability |
|---|
|---|---|---|
| Sum = 2 | (1,1) | 1/36 |
|---|---|---|
| Sum = 7 | (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) | 6/36 = 1/6 |
| Sum = 12 | (6,6) | 1/36 |
| Doubles | (1,1),(2,2),...,(6,6) | 6/36 = 1/6 |
#### Cards (Standard Deck = 52 cards)
Solved Examples
Example 1 — Basic
Q: A die is thrown. Find P(prime number).
Prime numbers on die: 2, 3, 5 → 3 favourable outcomes
Total outcomes: 6
P(prime) = 3/6 = 1/2
Example 2 — Cards
Q: A card is drawn from a well-shuffled deck. Find:
(a) P(red ace) (b) P(face card) (c) P(neither jack nor king)
(a) Red aces: Ace of Hearts, Ace of Diamonds = 2 → P = 2/52 = 1/26
(b) Face cards: 12 → P = 12/52 = 3/13
(c) Jacks = 4, Kings = 4, total to exclude = 8. Favourable = 52−8 = 44 → P = 44/52 = 11/13
Example 3 — Two Dice
Q: Two dice are thrown. Find P(sum ≤ 5).
List outcomes where sum ≤ 5:
Sum=2: (1,1) → 1
Sum=3: (1,2),(2,1) → 2
Sum=4: (1,3),(2,2),(3,1) → 3
Sum=5: (1,4),(2,3),(3,2),(4,1) → 4
Total favourable = 10
P = 10/36 = 5/18
Example 4 — Word Problem
Q: A bag contains 5 red, 3 black, and 2 white balls. One ball is drawn at random. Find P(not black).
Total = 10, Black = 3
P(not black) = 1 − 3/10 = 7/10
Or directly: non-black balls = 7 → P = 7/10 ✓
PYQs
2023
Q: A box contains 3 blue, 2 white, and 4 red marbles. A marble is drawn randomly. Find P(not red).
Total = 9, Red = 4, Not red = 5
P(not red) = 5/9
2022
Q: Two players A and B take turns. A wins if he gets a 6 on his throw. B wins if she gets a sum of 7 on two dice. Who has a better chance?
P(A wins) = 1/6
P(B wins) = 6/36 = 1/6
Equal chances
2021
Q: What is the probability that a leap year chosen at random will have 53 Sundays?
A leap year = 366 days = 52 weeks + 2 extra days
The 2 extra days can be: (Mon,Tue), (Tue,Wed), (Wed,Thu), (Thu,Fri), (Fri,Sat), (Sat,Sun), (Sun,Mon)
Total = 7 possibilities. Sundays in 53 → 2 cases include Sunday: (Sat,Sun) and (Sun,Mon)
P = 2/7
2020
Q: A card is drawn from a shuffled deck. What is the probability of drawing a card that is not a face card?
Face cards = 12 (J, Q, K of 4 suits)
Not face cards = 52 − 12 = 40
P = 40/52 = 10/13
MCQ Practice
Q1. P(impossible event) = (A) 1 (B) 0.5 (C) 0 ✓ (D) Undefined
Q2. From digits 1-9, probability of picking a perfect square:
Perfect squares: 1, 4, 9 → 3 numbers
P = 3/9 = 1/3 ✓
Q3 (Hard). Cards numbered 1 to 20 are shuffled. Find P(number is prime OR divisible by 3).
Primes (1-20): 2,3,5,7,11,13,17,19 → 8
Divisible by 3: 3,6,9,12,15,18 → 6
Both (prime AND div by 3): just 3 → 1
P = (8 + 6 − 1)/20 = 13/20
Revision Notes
Common Mistakes:
❌ Not listing ALL outcomes for two-dice problems (36, not 11)
❌ Counting HT and TH as the same (they're not!)
❌ Writing probability > 1 or < 0 (always check: 0 ≤ P ≤ 1)
❌ For "not" events, always use P(Ē) = 1 − P(E) instead of counting manually