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Agniveer (All) Probability

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This page covers Agniveer (All) Probability with complete concept notes, 38 graded practice MCQs, key points and exam-specific tips. Free to study.

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Concept Notes

Probability— Rules & Concept

1.Core Concept

Read this first — the foundation of the topic

Probability is the mathematical study of uncertainty and chance. It measures how likely an event is to occur. Think of it as a number between 0 and 1, where 0 means impossible and 1 means certain. Core Concept: Probability = Number of favorable outcomes / Total number of possible outcomes. For example, when flipping a coin, probability of heads = 1/2 = 0.5.

2.Key Rules

Core rules you must know cold
Addition Rule

P(A or B) = P(A) + P(B) - P(A and B) 2

Multiplication Rule

P(A and B) = P(A) × P(B) for independent events 3

Complement Rule

P(not A) = 1 - P(A) 4

Conditional Probability

P(A|B) = P(A and B) / P(B)

3.Formula Block

Memorise — at least one formula appears in every paper
• Basic Probability: P(E) = n(E) / n(S)
• Combination: nCr = n! / (r! × (n-r)!)
• Permutation: nPr = n! / (n-r)!
• Expected Value: E(X) = Σ x × P(x)

4.Exam Patterns

What examiners ask — read before attempting PYQs

NDA typically asks card problems, dice problems, bag and ball questions, and conditional probability. Most questions are 2-3 marks. Common formats include: 'What is the probability that...', 'Find the chance of...', 'If two events...'.

5.Shortcuts

Use these to save 30–60 seconds per question
Card Memory Trick

Total cards = 52, Red = 26, Black = 26, Face cards = 12, Aces = 4 2

Dice Sum Shortcut

For two dice, total outcomes = 36. Sum of 7 has maximum probability (6/36 = 1/6) 3. At Least One Formula: P(at least one) = 1 - P(none)

6.Worked Example

Solve this step-by-step before moving on
1
Step 1

Total balls = 5 + 3 = 8

2
Step 2

Ways to choose 2 balls from 8 = 8C2 = 28

3
Step 3

Ways to choose 2 red balls from 5 = 5C2 = 10

4
Step 4

Probability = 10/28 = 5/14 Worked Example 2: Two dice are thrown. Find probability that sum is greater than 8.

1
Step 1

Total outcomes = 6 × 6 = 36

2
Step 2

Favorable outcomes (sum > 8): Sum = 9: (3,6), (4,5), (5,4), (6,3) = 4 ways Sum = 10: (4,6), (5,5), (6,4) = 3 ways Sum = 11: (5,6), (6,5) = 2 ways Sum = 12: (6,6) = 1 way

3
Step 3

Total favorable = 4 + 3 + 2 + 1 = 10

4
Step 4

Probability = 10/36 = 5/18 Most

7.Exam Traps

Common mistakes students make — avoid these

(#1): Students confuse 'with replacement' and 'without replacement' problems. In without replacement, the sample space changes after each draw. Always check if items are put back or not.

This changes the denominator in subsequent calculations and can completely alter your answer.

Key Points to Remember

  • Probability always lies between 0 and 1 (inclusive)
  • P(E) = Favorable outcomes / Total outcomes
  • P(A) + P(not A) = 1 for any event A
  • For independent events: P(A and B) = P(A) × P(B)
  • Card shortcut: 52 total, 26 red, 26 black, 12 face cards, 4 aces
  • Dice shortcut: Two dice give 36 total outcomes, sum of 7 most probable
  • At least one formula: P(at least one) = 1 - P(none)
  • Conditional probability: P(A|B) = P(A and B) / P(B)
  • Combination nCr = n!/(r!(n-r)!) for selection problems
  • Always check if sampling is with or without replacement

Exam-Specific Tips

  • Standard deck has exactly 52 cards with 13 cards in each suit
  • Number of ways to arrange n objects = n! factorial
  • Probability of getting same number on two dice = 6/36 = 1/6
  • Maximum value of probability is 1 (certainty)
  • In conditional probability, P(A|B) means probability of A given B has occurred
  • Expected value formula: E(X) = Σ x × P(x)
  • For mutually exclusive events: P(A or B) = P(A) + P(B)
  • Binomial probability involves exactly r successes in n trials
Practice MCQs

Probability — Practice Questions

38graded MCQs · easy to hard · full solution & trap analysis · showing 20 of 38

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Practice 1easy

Let **a** = 2**i** + 3**j** + **k** and **b** = **i** − **j** + 2**k**. The scalar projection of **a** onto **b** is:

Practice 2easy

The direction cosines of the line passing through points P(1, 2, 3) and Q(4, 5, 6) are:

Practice 3easy

If **a** and **b** are two vectors such that |**a**| = 5, |**b**| = 3, and **a** · **b** = 12, then the angle θ between them is:

Practice 4easy

The vectors **a** = **i** + 2**j** + 3**k** and **b** = 2**i** + 4**j** + 6**k** are:

Practice 5easy

The equation of the plane passing through the point (1, 2, 3) and perpendicular to the vector **n** = 2**i** − **j** + 3**k** is:

Practice 6easy

Let **a** = 2**i** + 3**j** + **k** and **b** = **i** − **j** + 2**k**. The magnitude of the vector projection of **a** onto **b** is:

Practice 7easy

If **u** = **i** + 2**j** − **k** and **v** = 3**i** − **j** + 2**k**, then the magnitude of **u** × **v** is:

Practice 8easy

Two vectors **a** and **b** satisfy |**a**| = 5, |**b**| = 3, and **a** · **b** = 12. The angle θ between them is:

Practice 9easy

The scalar triple product (**a** × **b**) · **c** for **a** = **i** + **j**, **b** = **j** + **k**, and **c** = **k** + **i** is:

Practice 10easy

Let vectors **a** = (1, 2, 3), **b** = (2, -1, 1), and **c** = (1, 1, 1). The scalar triple product [**a** **b** **c**] = **a** · (**b** × **c**) is equal to:

Practice 11easy

Two vectors **u** and **v** satisfy |**u**| = 3, |**v**| = 4, and **u** · **v** = 6. The angle θ between **u** and **v** is:

Practice 12easy

A line passes through the point P(1, 2, 3) and has direction vector **d** = (2, −1, 2). The distance from the point Q(4, 3, 5) to this line is:

Practice 13easy

Three events A, B, and C are mutually exclusive and exhaustive with P(A) = 1/2, P(B) = 1/3, and P(C) = 1/6. If event D occurs with probability P(D|A) = 1/4, P(D|B) = 1/2, and P(D|C) = 1/3, then P(D) is:

Practice 14easy

A vector **v** makes angles α, β, and γ with the positive x, y, and z axes respectively. If cos α = 1/2 and cos β = 1/3, then cos γ is equal to:

Practice 15medium

Two vectors **a** = (1, 2, 3) and **b** = (2, −1, 1) are given. A third vector **c** is perpendicular to both **a** and **b**. If **c** · (1, 1, 1) = 5, then **c** is equal to:

Practice 16medium

Let **u** = (cos θ, sin θ, 0) and **v** = (sin θ, −cos θ, 1) be two vectors. The angle α between **u** and **v** satisfies cos α = k/√2, where k is a constant. The value of k is:

Practice 17medium

A plane passes through the point (1, 2, 3) and is perpendicular to the vector **n** = (2, −1, 2). The distance from the origin O(0, 0, 0) to this plane is:

Practice 18medium

In a random experiment, two events A and B are such that P(A) = 1/3, P(B) = 1/2, and P(A ∩ B) = 1/6. A vector **u** is defined as **u** = (P(A ∪ B), P(A' ∩ B), P(A ∩ B')), where A' and B' denote the complements of A and B. The magnitude of **u** is:

Practice 19medium

Two vectors **a** = (1, 2, 3) and **b** = (2, −1, 1) are given. A third vector **c** is perpendicular to both **a** and **b**. If **c** · (1, 1, 1) = 5, find **c**.

Practice 20medium

A line passes through the point P(1, 2, 3) and is parallel to the vector **d** = (2, −1, 2). The distance from the point Q(4, 5, 6) to this line is d_Q. Find d_Q.

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60-Second Revision — Probability

  • Remember: Probability = Favorable/Total, always between 0 and 1
  • Formula: P(at least one) = 1 - P(none) for complex problems
  • Trap: Check if sampling is with or without replacement
  • Cards: 52 total, 26 red, 12 face cards, 4 aces per suit
  • Dice: 36 total outcomes for two dice, sum 7 most probable
  • Independent events: Multiply probabilities P(A) × P(B)
  • Complement rule: P(not A) = 1 - P(A) saves calculation time
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