Probability Short Notes | Quick Revision for Competitive Exams

 PROBABILITY – COMPLETE SHORT NOTES (FOR COMPETITIVE EXAMS)

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1. Basic Definitions

Experiment

An action whose outcome cannot be predicted exactly.
(E.g., tossing a coin, rolling a die)

Sample Space (S)

Set of all possible outcomes.

• Coin → {H, T}

• Die → {1,2,3,4,5,6}

Event (E)

Subset of sample space.

• Getting even number in die = {2,4,6}

Favourable outcomes

Outcomes satisfying the condition.

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2. Classical Probability Formula

$$P(E)=\frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}$$

Properties

• $0\le P(E)\le 1$

• Impossible event → P = 0

• Certain event → P = 1

• Sum of probabilities of all elementary events = 1

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3. Complementary Probability

$$P(\text{not E})=1-P(E)$$

Useful for:

• At least one type questions

• Non-occurrence type questions

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4. Mutually Exclusive Events

Events that cannot occur together.

$$P(A\text{ or }B)=P(A)+P(B)$$

Example: Drawing either king or queen.

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5. Non-Mutually Exclusive Events

$$P(A\text{ or }B)=P(A)+P(B)-P(A\cap B)$$

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6. Independent Events

Events that do not affect each other.

$$P(A\cap B)=P(A)P(B)$$

Examples: Coin toss + Rolling a die, 2 bulbs picked with replacement.

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7. Dependent Events

Without replacement cases.

$$P(A\cap B)=P(A)\times P(B\mathrm{\mid }A)$$

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8. Conditional Probability

$$P(A\mathrm{\mid }B)=\frac{P(A\cap B)}{P(B)}$$

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9. Odds

• Odds in favour = Favourable : Unfavourable

• Odds against = Unfavourable : Favourable

• Conversion:
  If odds in favour = a : b →

$$P(E)=\frac{a}{a+\mathrm{b}}$$

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10. Important Standard Results

Coin Toss

• n coins → $2^n$outcomes

• Probability of exactly k head

$$P = \frac{{^nC_k}}{2^n}$$

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Die

• Probability of multiple rolls:

  • Sum, even, odd, multiples etc.

• P(getting number > 2) = 4/6 = 2/3

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Cards (52 cards)

• Hearts, Diamonds = Red → 26

• Spades, Clubs = Black → 26

• Face cards = 12

• Each suit → 13 cards

Common probabilities:

• P(red) = 26/52 = 1/2

• P(face card) = 12/52 = 3/13

• P(ace) = 4/52 = 1/13

Without replacement → dependent
With replacement → independent

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11. At Least One Formula

$$P(\text{At least one})=1-P(\text{None})$$

Example: Probability of at least 1 success in n trials.

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12. Bayes’ Theorem

If events (A_1, A_2, A_3 … A_n) form a partition:

$$P(A_i\mathrm{\mid }B)=\frac{P(A_i)\cdot P(B\mathrm{\mid }{A}_i)}{\sum P(A_j)P(B\mathrm{\mid }{A}_j)}$$

Used in:

• Test accuracy

• Selection problems

• Defective items

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13. Probability Using Permutation & Combination

$$P(E)=\frac{{}^n{C}_k\text{ (favourable)}}{{}^n{C}_k\text{ (total)}}$$

Used for:

• Seating

• Committee selection

• Arrangements

• Card combinations

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14. Geometric Probability

$$P(E)=\frac{\text{Length/Area/Volume of favourable region}}{\text{Total region}}$$

Used in:

• Points on line segment

• Random chord problems

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15. High-Scoring Tricks (Exam Special)

1. Replacement = independent

• Probability remains same for each draw.

2. Without replacement = dependent

• Denominator decreases each step.

3. "Either-or" keyword → use P(A) + P(B)

• If overlap exists → subtract intersection.

4. "Both" keyword

• Multiply probabilities.

5. “At least one” → use complement

• Always faster.

6. Probability never > 1

• Good quick check for errors.

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16. One-Liners for Fast Revision

• Total outcomes = product of individual outcomes.

• Probability of success + failure = 1.

• If P(A) + P(B) > 1 → events overlap.

• Picking 2 cards from 52 is combination-based, not simple probability.

• For independent events: variance increases; probability multiplies.

• For mutually exclusive: intersection = 0.

• Probability of equal likelihood outcomes uses classical formula.