Important Formulas & Concepts

Probability

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Probability

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Important Formulas & Concepts

Study Material

Probability

Probability is one of the most important chapters in Quantitative Aptitude and Mathematics. It deals with the likelihood or chance of occurrence of an event. Questions from Probability are frequently asked in SSC, Banking, Railway, CAT, NDA, CDS, Insurance, Defence, and various competitive examinations.

This chapter mainly deals with:

  • Random experiments
  • Sample space and events
  • Coin and dice problems
  • Playing card probability
  • Independent and dependent events
  • Mutually exclusive events
  • Combination-based probability
  • Conditional probability concepts

Understanding probability concepts and formulas helps candidates solve aptitude questions quickly and accurately.

What is Probability?

Probability means the chance or likelihood of occurrence of an event.

It measures how likely an event is to happen.

Examples:

  • Getting Head when a coin is tossed
  • Getting number 6 on a die
  • Drawing an Ace from playing cards

Probability = Number of Favourable Outcomes / Total Number of Possible Outcomes

Basic Probability Formula

If:

  • Total possible outcomes = n(S)
  • Favourable outcomes = n(E)

then:

P(E) = n(E) / n(S)

Important Terms in Probability

1. Experiment

An operation that produces well-defined outcomes is called an Experiment.

Examples:

  • Tossing a coin
  • Rolling a die
  • Drawing a card

2. Random Experiment

An experiment whose exact result cannot be predicted in advance is called a Random Experiment.

Examples:

  • Tossing a fair coin
  • Rolling an unbiased die
  • Drawing a card from shuffled cards

3. Sample Space

The set of all possible outcomes is called the Sample Space.

It is generally denoted by S.

Experiment Sample Space
Tossing a Coin {H, T}
Throwing a Die {1,2,3,4,5,6}
Tossing Two Coins {HH, HT, TH, TT}

4. Event

Any subset of a sample space is called an Event.

Example:

Getting an even number while rolling a die:

E = {2,4,6}

5. Favourable Outcomes

Outcomes that satisfy the required condition are called favourable outcomes.

Important Probability Rules

1. Probability Range

0 ≀ P(E) ≀ 1

  • P(E) = 0 β†’ Impossible Event
  • P(E) = 1 β†’ Sure Event

2. Probability of Sure Event

P(S) = 1

3. Probability of Impossible Event

P(Ξ¦) = 0

4. Complementary Probability

If A is an event:

P(Not A) = 1 βˆ’ P(A)

Types of Events

1. Independent Events

Two events are independent if occurrence of one does not affect the other.

Example:

  • Tossing a coin twice
  • Rolling two dice

For independent events:

P(A and B) = P(A) Γ— P(B)

2. Dependent Events

Two events are dependent if occurrence of one affects the other.

Example:

Drawing two cards without replacement.

3. Mutually Exclusive Events

Two events that cannot occur simultaneously are called mutually exclusive events.

Example:

  • Getting Head and Tail together
  • Getting 2 and 5 together on one die

For mutually exclusive events:

P(A or B) = P(A) + P(B)

Addition Rule of Probability

For any two events A and B:

P(A βˆͺ B) = P(A) + P(B) βˆ’ P(A ∩ B)

Multiplication Rule of Probability

For independent events:

P(A and B) = P(A) Γ— P(B)

Probability of Coin Toss

Event Probability
Getting Head 1/2
Getting Tail 1/2

Probability of Dice Throw

Event Probability
Getting 1 1/6
Getting Even Number 3/6 = 1/2
Getting Prime Number 3/6 = 1/2

Playing Cards Concepts

A standard deck contains:

  • 52 cards
  • 4 suits
  • 13 cards in each suit

Types of Suits

Suit Color
Spades Black
Clubs Black
Hearts Red
Diamonds Red

Important Card Facts

  • Face Cards = King, Queen, Jack
  • Total Face Cards = 12
  • Total Kings = 4
  • Total Queens = 4
  • Total Jacks = 4
  • Total Aces = 4

Combination Formula Used in Probability

Many probability questions use combinations.

nCr = n! / [r!(n-r)!]

Important Concepts for Exams

1. At Least One

Use complementary probability:

P(At least one) = 1 βˆ’ P(None)

2. Without Replacement

Probability changes after every draw.

Events become dependent.

3. With Replacement

Probability remains same after every draw.

Events remain independent.

Quick Revision Formula Table

Concept Formula
Basic Probability n(E)/n(S)
Complementary Probability 1 βˆ’ P(A)
Independent Events P(A) Γ— P(B)
Mutually Exclusive Events P(A) + P(B)
Addition Rule P(AβˆͺB)
Combination Formula nCr

Common Mistakes to Avoid

  • Ignoring total sample space.
  • Confusing dependent and independent events.
  • Using wrong denominator.
  • Ignoring replacement condition.
  • Calculation mistakes in combinations.

Important Exam Tips

  • Always find total outcomes first.
  • Use complementary probability for β€œat least” problems.
  • Learn playing card concepts thoroughly.
  • Practice dice and coin problems regularly.
  • Memorize important probability formulas.
  • Use combinations carefully.
  • Read question conditions properly.

Probability is an important aptitude topic that combines logic, arithmetic, and analytical thinking. Strong understanding of formulas, events, and probability rules helps candidates solve competitive examination questions quickly and accurately.

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