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Elementary Probability

Quantitative Aptitude · Statistics, Probability & Data Skills

Useful for: SSC CGL, UPSC NDA/NA, CSIR NET

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Overview

Probability = favourable outcomes ÷ total equally likely outcomes; complement and counting rules handle most items.

Key points

P(A) lies between 0 and 1; P(A) + P(not A) = 1.
Two dice: 36 outcomes; two coins: 4; three coins: 8.
A pack of cards: 52 cards, 4 suits of 13, 26 red and 26 black, 12 face cards, 4 aces.
For independent events, P(A and B) = P(A) × P(B).
"At least one" = 1 − P(none).

Explanation

Sum 7 on two dice is the most likely total (6 ways).
Mutually exclusive events: P(A or B) = P(A) + P(B).
Without replacement, the total reduces after each draw.
Odds in favour a:b means P = a/(a + b).

Step by step

Define the sample space.
Count favourable cases.
Divide and simplify.

Examples

Two dice, sum 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) → 6/36 = 1/6.
A red king from a pack: 2/52 = 1/26.
At least one head in two tosses: 1 − 1/4 = 3/4.

Traps and advanced points

Trap: Counting (2,5) and (5,2) as one outcome for two distinguishable dice.
Trap: Forgetting that face cards are 12 (J, Q, K of four suits).
Trap: Adding probabilities of events that can happen together.

Quick revision

P(not A) = 1 − P(A).
Dice 36, cards 52.
At least one = 1 − none.

Practice MCQs

From the perspective of definition and scope, which one of the following statements about Probability is correct?
1. Do not apply a memorized Probability formula without checking its conditions.
2. Probability is unrelated to institutions, processes or real-world applications.
3. Probability has no exceptions or limitations in exam analysis.
4. Probability questions reward a formula-first approach and accurate algebraic manipulation.
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Answer: (d) Probability questions reward a formula-first approach and accurate algebraic manipulation.. Probability questions reward a formula-first approach and accurate algebraic manipulation.
Consider the following statements on Probability, focusing on definition and scope: 1. Probability questions reward a formula-first approach and accurate algebraic manipulation. 2. Advanced questions often combine Probability with another algebraic or geometric idea. Which of the statements given above is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
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Answer: (c) Both 1 and 2. Both statements are correct.
Which option best connects the definition and scope dimension of Probability with its broader exam relevance?
1. For Probability, write the governing relation before substituting numbers. Also, Check domains, signs and units before finalizing the option.
2. Do not apply a memorized Probability formula without checking its conditions.
3. Every statement about Probability is absolute and context-free.
4. Probability can be understood without any mechanism, limitation or exception.
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Answer: (a) For Probability, write the governing relation before substituting numbers. Also, Check domains, signs and units before finalizing the option.. For Probability, write the governing relation before substituting numbers. Also, Check domains, signs and units before finalizing the option.
A bag has 2 red and 13 blue balls. One ball is drawn at random. Probability of red?
1. 2/15
2. 2/13
3. 1/15
4. 13/15
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Answer: (a) 2/15. Favourable outcomes=2; total=15; probability=2/15.
A bag has 7 red and 15 blue balls. One ball is drawn at random. Probability of red?
1. 7/15
2. 7/22
3. 15/22
4. 1/22
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Answer: (b) 7/22. Favourable outcomes=7; total=22; probability=7/22.
A bag has 8 red and 8 blue balls. One ball is drawn at random. Probability of red?
1. 1/16
2. 8/8
3. Alternative 1
4. 8/16
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Answer: (d) 8/16. Favourable outcomes=8; total=16; probability=8/16.
A bag has 7 red and 13 blue balls. One ball is drawn at random. Probability of red?
1. 7/13
2. 13/20
3. 1/20
4. 7/20
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Answer: (d) 7/20. Favourable outcomes=7; total=20; probability=7/20.
A bag has 8 red and 7 blue balls. One ball is drawn at random. Probability of red?
1. 1/15
2. 8/15
3. 8/7
4. 7/15
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Answer: (b) 8/15. Favourable outcomes=8; total=15; probability=8/15.
A bag has 8 red and 6 blue balls. One ball is drawn at random. Probability of red?
1. 6/14
2. 1/14
3. 8/14
4. 8/6
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Answer: (c) 8/14. Favourable outcomes=8; total=14; probability=8/14.
A bag has 7 red and 12 blue balls. One ball is drawn at random. Probability of red?
1. 1/19
2. 7/19
3. 7/12
4. 12/19
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Answer: (b) 7/19. Favourable outcomes=7; total=19; probability=7/19.