Quick Concept Summary
Probability measures how likely an event is to occur. It always lies between 0 (impossible) and 1 (certain).
Core formula:
Key vocabulary:
| Term | Meaning |
|---|---|
| Experiment | An action with uncertain outcomes (tossing a coin) |
| Sample Space (S) | All possible outcomes |
| Event (E) | A specific set of outcomes we care about |
| Favourable outcome | Outcome that satisfies the event condition |
Types of events:
- Certain event: P = 1 (always happens)
- Impossible event: P = 0 (never happens)
- Complementary event: P(A') = 1 − P(A)
- Mutually exclusive: Two events that cannot both happen (P(A∩B) = 0)
- Independent: One event does not affect the other
Key Formulas
| Rule | Formula | When to use | |
|---|---|---|---|
| Basic probability | P(E) = n(E)/n(S) | Any single event | |
| Complement | P(A') = 1 − P(A) | "At least one", "not" problems | |
| Addition (mutually exclusive) | P(A∪B) = P(A) + P(B) | A and B can't happen together | |
| Addition (general) | P(A∪B) = P(A) + P(B) − P(A∩B) | A and B may overlap | |
| Multiplication (independent) | P(A∩B) = P(A) × P(B) | Separate experiments | |
| Multiplication (dependent) | P(A∩B) = P(A) × P(B\ | A) | Drawing without replacement |
Worked Examples
Example 1 — Coin Toss (Basic)
Q: A coin is tossed. What is the probability of getting a Head?
- Sample space S =
{H, T}→ n(S) = 2 - Favourable outcomes (Head) =
{H}→ n(E) = 1 - P(Head) = 1/2 = 0.5
Example 2 — Single Die (Basic)
Q: A die is rolled once. Find the probability of getting a number greater than 4.
- S =
{1, 2, 3, 4, 5, 6}→ n(S) = 6 - Numbers greater than 4:
{5, 6}→ n(E) = 2 - P(>4) = 2/6 = 1/3
Example 3 — Card Draw (Intermediate)
Q: A card is drawn from a standard 52-card deck. Find the probability it is a King or a Heart.
- P(King) = 4/52
- P(Heart) = 13/52
- P(King AND Heart) = 1/52 ← King of Hearts
P(King OR Heart) = 4/52 + 13/52 − 1/52 = 16/52 = 4/13
Example 4 — Balls in a Bag (Complementary)
Q: A bag contains 3 red, 4 blue, and 5 green balls. A ball is drawn at random. Find the probability that it is NOT red.
- Total balls = 3 + 4 + 5 = 12
- P(Red) = 3/12 = 1/4
- P(NOT Red) = 1 − 1/4 = 3/4
Alternatively: Non-red balls = 4 + 5 = 9 → P = 9/12 = 3/4 ✓
Example 5 — Two Independent Events (Multiplication)
Q: A coin is tossed and a die is rolled simultaneously. Find the probability of getting a Head and an even number.
- P(Head) = 1/2
- P(Even number) = P(
{2,4,6}) = 3/6 = 1/2
P(Head AND Even) = 1/2 × 1/2 = 1/4
Example 6 — Drawing Without Replacement (Dependent)
Q: Two cards are drawn one after another from a 52-card deck without replacement. Find the probability both are Aces.
- P(1st card is Ace) = 4/52
- P(2nd card is Ace | 1st was Ace) = 3/51 ← only 3 Aces left in 51 cards
P(Both Aces) = 4/52 × 3/51 = 12/2652 = 1/221
Shortcuts and Tricks
TNPSC / Placement Exam Pattern
TNPSC: 1–2 questions per paper, usually from coins, dice, or coloured balls. Difficulty: basic to intermediate. Rarely goes beyond single-step problems.
Placement (TCS, Infosys, Wipro): 2–4 questions, may include cards, dependent events (without replacement), and "at least one" type using complements.
Common question types: 1. Single event — coin / die / card / bag 2. Complementary ("NOT", "at least one") 3. Two independent events (coin + die together) 4. Dependent events (drawing without replacement) 5. Sum of two dice
Practice Questions
| # | Question | Answer |
|---|---|---|
| 1 | A die is rolled. P(getting 3 or 5)? | 2/6 = 1/3 |
| 2 | Two coins tossed. P(exactly one Head)? | 2/4 = 1/2 |
| 3 | Bag: 5 red, 3 white balls. P(white)? | 3/8 |
| 4 | Card drawn from 52 cards. P(face card)? | 12/52 = 3/13 |
| 5 | Die rolled. P(even number)? | 3/6 = 1/2 |
| 6 | Three coins tossed. P(all tails)? | 1/8 |
| 7 | Bag: 4 red, 6 blue. P(NOT blue)? | 4/10 = 2/5 |
| 8 | Card drawn. P(Heart or Ace)? | 16/52 = 4/13 |
| 9 | Two dice rolled. P(sum = 7)? | 6/36 = 1/6 |
| 10 | Two cards drawn without replacement. P(both red)? | 26/52 × 25/51 = 25/102 |