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NDA Probability

Study Material — 5 PYQs (2018–2020) · Concept Notes · Shortcuts

NDA Probability is a frequently tested subtopic — 5 previous year questions from 2018–2020 papers are included below with concept notes, key rules and shortcut tricks.

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2018–2020

13 Practice

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10 Key Points

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Previous Year Questions

NDA Probability — Past Exam Questions

5 questions from actual NDA papers · all shown free · click option to reveal solution

Exam Q 12018Previous Year Pattern

A bag contains 5 red balls, 3 blue balls, and 2 green balls. If one ball is drawn at random from the bag, what is the probability that it is neither red nor green?

Exam Q 22020Previous Year Pattern

A bag contains 5 red balls, 3 blue balls, and 2 green balls. If one ball is drawn at random from the bag, what is the probability that it is neither red nor green?

Exam Q 32020Previous Year Pattern

A bag contains 5 red balls, 7 blue balls, and 8 green balls. Two balls are drawn simultaneously at random from the bag. What is the probability that both balls are of different colours?

Exam Q 42018Previous Year Pattern

A bag contains 4 red balls, 5 blue balls, and 3 green balls. Two balls are drawn at random without replacement. What is the probability that both balls are of the same colour?

Exam Q 52019Previous Year Pattern

A bag contains 5 red balls, 7 blue balls, and 8 green balls. Two balls are drawn simultaneously at random from the bag. What is the probability that both balls are of different colours?

Concept Notes

Probability— Rules & Concept

Core ConceptRead this first — the foundation of the topic

PROBABILITY — CORE CONCEPT Probability tells us how likely an event is to happen. Think of it as a number between 0 and 1. If something is impossible, probability = 0. If something is certain, probability = 1. Everything else falls in between.

Simple definition: Probability of an event = (Number of favourable outcomes) / (Total number of possible outcomes) P(E) = n(E) / n(S)

Where n(E) = number of ways the event can happen, n(S) = total outcomes in the sample space. ---

Key RulesCore rules you must know cold

0 ≤ P(E) ≤ 1 always. Probability is never negative and never more than 1.

P(E) + P(E') = 1, where E' means the event does NOT happen. So P(E') = 1 - P(E).

If two events cannot happen at the same time, they are called Mutually Exclusive Events. P(A or B) = P(A) + P(B).

If two events do not affect each other, they are called Independent Events. P(A and B) = P(A) × P(B).

P(A or B) = P(A) + P(B) - P(A and B) — this is the Addition Rule.

Formula BlockMemorise — at least one formula appears in every paper

Basic Formula: P(E) = Favourable Outcomes / Total Outcomes

Complement Rule: P(E') = 1 - P(E)

Addition Rule: P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

Multiplication Rule (Independent): P(A ∩ B) = P(A) × P(B)

Odds in Favour: P(E) / P(E') = Favourable / Unfavourable

Odds Against: P(E') / P(E) = Unfavourable / Favourable

---

Exam PatternsWhat examiners ask — read before attempting PYQs

— WHAT GETS ASKED SSC CGL tests probability in 3 main ways: 1. Coin, Dice, and Card problems — most common. 2. Balls from a bag (red, blue, green balls drawn randomly). 3. Two-event problems using AND/OR logic. For cards: A standard deck has 52 cards — 4 suits (Hearts, Diamonds, Clubs, Spades), 13 cards each.

Each suit has Ace, 2 to 10, Jack, Queen, King. Face cards = Jack, Queen, King = 12 total. Red cards = 26 (Hearts + Diamonds). Black cards = 26 (Clubs + Spades). For dice: One die has 6 faces (1 to 6).

Two dice give 6 × 6 = 36 total outcomes. For coins: One coin gives 2 outcomes (H or T). Two coins give 4 outcomes. Three coins give 8 outcomes. ---

ShortcutsUse these to save 30–60 seconds per question

Trick 1 — Complement Shortcut

Instead of counting what you WANT, count what you DON'T want. P(at least one) = 1 - P(none). This saves massive time in exam

Trick 2 — Odds to Probability

If odds in favour = m : n, then P(E) = m / (m + n). If odds against = m : n, then P(E) = n / (m + n). Trick 3 — Multiply for AND, Add for OR (only if mutually exclusive): 'AND' means multiply the probabilities. 'OR' means add the probabilities (when events cannot happen together). ---

Worked ExampleSolve this step-by-step before moving on

1

Step 1

Total outcomes = 6 × 6 = 36

2

Step 2

Pairs that give sum 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6 pairs

3

Step 3

P(sum = 7) = 6/36 = 1/6 Answer: 1/6 --- WORKED EXAMPLE 2 — Balls in a Bag (At Least One) Question: A bag has 4 red and 6 blue balls. Two balls are drawn randomly. What is the probability that at least one is red?

1

Step 1

Total balls = 10. Total ways to pick 2 = 10C2 = 45

2

Step 2

Use complement. P(at least one red) = 1 - P(no red ball)

3

Step 3

P(no red) = ways to pick 2 blue balls = 6C2 = 15

4

Step 4

P(no red) = 15/45 = 1/3

5

Step 5

P(at least one red) = 1 - 1/3 = 2/3 Answer: 2/3 ---

Exam TrapsCommon mistakes students make — avoid these

— THE NUMBER ONE TRAP Students confuse 'at least one' problems. They try to count all cases individually (one red + one blue, two red) and often miss combinations. ALWAYS use the complement method: P(at least one) = 1 - P(none).

It is faster and error-free. Also, students forget that drawing cards or balls without replacement changes the total for the second draw — always check if replacement is mentioned.

Key Points to Remember

P(E) = Favourable Outcomes / Total Outcomes — this is the base formula for all problems.
Probability always lies between 0 and 1. It can never be negative or greater than 1.
Complement Rule: P(E') = 1 - P(E). Use this for 'at least one' problems — it saves time.
For AND of independent events: P(A and B) = P(A) × P(B). Multiply the probabilities.
For OR of mutually exclusive events: P(A or B) = P(A) + P(B). Add the probabilities.
A standard deck has 52 cards — 26 red, 26 black, 12 face cards, 4 Aces.
Two dice give 36 total outcomes. Sum of 7 has the highest probability = 6/36 = 1/6.
Odds in favour = m : n means P(E) = m / (m + n). Memorise this conversion formula.
Three coins tossed together give 8 total outcomes (2 raised to power 3).
Mutually exclusive events CANNOT occur together. Exhaustive events COVER all outcomes.

Exam-Specific Tips

A standard deck of playing cards has exactly 52 cards divided into 4 suits of 13 cards each.
The number of face cards in a standard deck is 12 (Jack, Queen, King across 4 suits).
When two dice are rolled, the total number of outcomes is 36 and sum of 7 is the most probable outcome with probability 1/6.
For n coins tossed simultaneously, total outcomes = 2^n. For 3 coins, total = 8.
If odds in favour of an event are a : b, then Probability = a / (a + b).
P(A ∪ B) = P(A) + P(B) - P(A ∩ B) is the General Addition Rule for any two events.
For mutually exclusive events, P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B).
The probability of getting at least one Head when 3 coins are tossed = 1 - (1/8) = 7/8.

Practice MCQs

Probability — Practice Questions

13graded MCQs · easy to hard · full solution & trap analysis

All MCQs →

Practice 1easy

A bag contains 5 red balls, 3 blue balls, and 2 green balls. If one ball is drawn at random, what is the probability that it is either red or green?

Practice 2easy

A coin is tossed 3 times. What is the probability of getting exactly 2 heads?

Practice 3easy

A coin is flipped 3 times. What is the probability of getting exactly 2 heads?

Practice 4easy

A bag contains 5 red balls, 3 blue balls, and 2 green balls. If one ball is drawn at random, what is the probability that it is either red or green?

Practice 5medium

A student has a 60% chance of passing Mathematics and a 75% chance of passing English. Assuming independence, what is the probability that the student passes at least one subject?

Practice 6medium

A bag contains 5 red balls, 7 blue balls, and 8 green balls. If two balls are drawn simultaneously at random, what is the probability that both balls are of the same colour?

Practice 7medium

In a lottery, the probability of winning a prize is 0.15. If 200 tickets are sold and a person buys 8 tickets, what is the probability that this person wins exactly 2 prizes?

Practice 8medium

A bag contains 5 red balls, 7 blue balls, and 8 green balls. If two balls are drawn simultaneously at random, what is the probability that both balls are of the same colour?

Practice 9medium

In a lottery, the probability of winning a prize is 0.12. If 500 tickets are sold, how many tickets are expected to win a prize?

Practice 10medium

A box contains 6 defective and 14 non-defective items. If 3 items are drawn at random without replacement, what is the probability that all 3 are non-defective?

Practice 11hard

A bag contains 5 red balls, 7 blue balls, and 8 green balls. Two balls are drawn simultaneously without replacement. What is the probability that both balls are of different colours?

Practice 12hard

A bag contains 5 red balls, 7 blue balls, and 8 green balls. Two balls are drawn simultaneously without replacement. What is the probability that both balls are of different colours?

Practice 13hard

Three fair dice are rolled simultaneously. What is the probability that the sum of the numbers shown is 10, given that at least one die shows a 4?

60-Second Revision — Probability

Formula: P(E) = Favourable Outcomes / Total Outcomes. Always write this first in any problem.
Trick: For 'at least one' problems, always use P(at least one) = 1 - P(none). Never count cases one by one.
Remember: AND means MULTIPLY (independent events). OR means ADD (mutually exclusive events).
Cards: 52 total, 26 red, 26 black, 12 face cards, 4 Aces. Memorise these before the exam.
Trap: Check if balls or cards are drawn WITH or WITHOUT replacement — it changes the total outcomes for the second draw.
Odds to Probability: Odds in favour m:n → P(E) = m/(m+n). Odds against m:n → P(E) = n/(m+n).
Complement Rule: P(E') = 1 - P(E). If P(E) = 3/7, then P(E') = 4/7. Always double-check they add to 1.

←Standard Deviation & Variance Permutation & Combination→

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