Core ConceptRead this first — the foundation of the topic
Probability is the mathematical way to measure how likely an event is to happen. Think of it as a fraction that tells you the chances of something occurring. In SSC CGL, probability questions appear in almost every paper and can fetch you easy marks if you master the basic concepts and shortcuts. Core Concept: Probability = Favorable Outcomes / Total Possible Outcomes. The value always lies between 0 and 1. If probability is 0, the event will never happen. If it's 1, the event will definitely happen.
Key RulesCore rules you must know cold
Addition Rule
For mutually exclusive events A and B, P(A or B) = P(A) + P(B)
2
Multiplication Rule
For independent events A and B, P(A and B) = P(A) × P(B)
3
Complementary Events
P(A) + P(not A) = 1, so P(not A) = 1 - P(A)
4
Conditional Probability
P(A given B) = P(A and B) / P(B)
Formula BlockMemorise — at least one formula appears in every paper
• Basic Probability = Favorable outcomes / Total outcomes
• Probability of at least one = 1 - Probability of none
• Odds in favor = Favorable : Unfavorable
• Expected value = Sum of (Probability × Value)
Exam PatternsWhat examiners ask — read before attempting PYQs
SSC CGL typically asks about card problems, dice problems, ball drawing from bags, and coin tossing. Questions often involve finding probability of getting specific combinations or calculating odds.
ShortcutsUse these to save 30–60 seconds per question
#1 - For 'At Least One' Problems: Instead of calculating all possible combinations, use the complement rule. Find the probability that the event doesn't happen at all, then subtract from 1.
Worked ExampleSolve this step-by-step before moving on
1
Step 1
Count total balls = 5 + 3 + 2 = 10
2
Step 2
Count favorable outcomes (red balls) = 5
3
Step 3
Apply formula = 5/10 = 1/2 = 0.5
Answer: 1/2 or 50%
Shortcut Trick #2 - Card Problems Quick Reference: Remember that a deck has 52 cards with 13 cards of each suit (hearts, diamonds, clubs, spades). Face cards are 12 in total (4 each of Jack, Queen, King).
Worked Example 2: Two dice are thrown. What is the probability of getting a sum of 7?
1
Step 1
Total possible outcomes when throwing two dice = 6 × 6 = 36
2
Step 2
Find favorable outcomes for sum = 7
(1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6 outcomes
3
Step 3
Apply formula = 6/36 = 1/6
Answer: 1/6
Shortcut Trick #3 - Dice Sum Patterns: For two dice, sum of 7 has maximum probability (6/36), while sums of 2 and 12 have minimum probability (1/36 each).
Exam TrapsCommon mistakes students make — avoid these
#1: Students often forget to consider all possible arrangements. For example, when drawing 2 balls from a bag, they count (Red, Blue) but forget (Blue, Red) as a separate case when order matters. Always check whether the problem requires considering order or not.
Another frequent error is mixing up 'with replacement' and 'without replacement' scenarios.
In 'with replacement', the total number of items remains the same for each draw. In 'without replacement', the total decreases after each draw, affecting subsequent probabilities.
Key Points to Remember
Probability always lies between 0 and 1 inclusive
Basic formula: Probability = Favorable outcomes / Total outcomes
For 'at least one' problems, use complement: P(at least one) = 1 - P(none)
Standard deck has 52 cards, 13 per suit, 12 face cards total
Two dice have 36 total outcomes, sum of 7 has highest probability
Mutually exclusive events: P(A or B) = P(A) + P(B)
Independent events: P(A and B) = P(A) × P(B)
Complementary events: P(A) + P(not A) = 1
Always check if order matters in the problem statement
With replacement vs without replacement changes denominators
Exam-Specific Tips
A standard deck contains exactly 52 cards with 4 suits of 13 cards each
Number of face cards in a deck is 12 (4 Jacks, 4 Queens, 4 Kings)
Two dice thrown together give 36 total possible outcomes
Sum of 7 with two dice can occur in exactly 6 ways
Probability of getting heads in a fair coin toss is exactly 1/2
In a standard deck, probability of drawing an ace is 4/52 = 1/13
Maximum probability value is 1, minimum probability value is 0
Number of ways to arrange n distinct objects is n! factorial
Practice MCQs
Probability — Practice Questions
6graded MCQs · easy to hard · full solution & trap analysis
A coin is tossed 3 times. What is the probability of getting exactly 2 heads?
Practice 2easy
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 3medium
In a class of 40 students, 15 play cricket, 20 play football, and 8 play both. If a student is selected at random, what is the probability that the student plays at least one sport?
Practice 4medium
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 5hard
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 6hard
In a lottery, 6 tickets are drawn from 100 tickets numbered 1 to 100. What is the probability that the sum of the numbers on the drawn tickets is even?
60-Second Revision — Probability
Formula: Probability = Favorable outcomes / Total outcomes
Remember: Use 1 - P(none) for 'at least one' questions
Trap: Don't confuse with replacement and without replacement scenarios
Quick fact: Standard deck = 52 cards, Face cards = 12
Pattern: Two dice sum of 7 has maximum probability of 1/6
Rule: Always check if order matters in arrangement problems
Shortcut: P(not A) = 1 - P(A) for complement problems