Probability - Aptitude Questions & Answers
Probability is one of the most important topics in Quantitative Aptitude. In this lesson, you will learn concepts, formulas, shortcuts, solved examples, and aptitude questions with answers. This topic is useful for exams like SSC, Bank, CAT, TCS, and other competitive exams.
Master probability concepts for aptitude tests with this comprehensive guide covering coins, dice, cards, and real-world applications. Learn formulas, shortcuts, and problem-solving techniques to calculate likelihood and chance in competitive exams.
Probability: Complete Guide with Formulas & Examples
What is Probability?
Probability measures how likely an event is to occur. It\'s calculated as the ratio of favorable outcomes to total possible outcomes.
Key Properties
- Probability always ranges between 0 and 1
- 0 = Event cannot happen (impossible)
- 1 = Event is certain to happen
- Probability can also be expressed as percentage (0% to 100%)
Four Main Categories
- Probability based on Coins
- Probability based on Dice
- Probability based on Cards
- Miscellaneous probability problems
Probability Based on Coins
Basic Coin Properties
- A coin has 2 faces: Head (H) and Tail (T)
- Total outcomes when tossing n coins = 2n
One Coin Toss
Total outcomes = 21 = 2 {H, T}
Two Coins Toss
Total outcomes = 22 = 4 {HH, HT, TH, TT}
Example: Two coins tossed simultaneously. Find probability of:
A) Identical toss (both same)
- Total outcomes = 4 {HH, HT, TH, TT}
- Favorable outcomes = {HH, TT} = 2
- Probability = 2/4 = 1/2 or 0.5
B) Two heads
- Favorable outcomes = {HH} = 1
- Probability = 1/4 = 0.25 or 25%
Three Coins Toss
Total outcomes = 23 = 8 {TTT, TTH, THT, HTT, THH, HTH, HHT, HHH}
Example: Three coins tossed simultaneously. Find probability of:
A) Two tails and one head
- Total outcomes = 8
- Favorable outcomes = {TTH, THT, HTT} = 3
- Probability = 3/8 = 0.375
B) At least one tail
- \"At least one tail\" means 1, 2, or 3 tails
- Favorable outcomes = All except {HHH} = 7
- Probability = 7/8 = 0.875 or 87.5%
Probability Based on Dice
Basic Dice Properties
- A standard die has 6 faces (1 to 6)
- Total outcomes when rolling n dice = 6n
Single Die Roll
Total outcomes = 61 = 6 {1, 2, 3, 4, 5, 6}
Example: Probability of getting 5 on a die roll = 1/6
Two Dice Roll
Total outcomes = 62 = 36
Example: Two dice rolled simultaneously. Find probability of:
A) Sum equals 4
- Favorable outcomes = {(1,3), (2,2), (3,1)} = 3
- Probability = 3/36 = 1/12
Shortcut Method: Two Dice Triangle
Memorize this pattern for sum probabilities:
Sum 3 → 2 ways
Sum 4 → 3 ways
Sum 5 → 4 ways
Sum 6 → 5 ways
Sum 7 → 6 ways
Sum 8 → 5 ways
Sum 9 → 4 ways
Sum 10 → 3 ways
Sum 11 → 2 ways
Sum 12 → 1 way (6,6)
For sum 4: 3 ways → Probability = 3/36 = 1/12
B) Sum is a multiple of 4
- Multiples of 4: 4, 8, 12
- Ways for sum 4 = 3
- Ways for sum 8 = 5
- Ways for sum 12 = 1
- Total favorable = 3 + 5 + 1 = 9
- Probability = 9/36 = 1/4
Three Dice Roll
Total outcomes = 63 = 216
Example: Three dice rolled. Probability of sum = 16?
Solution:
- Total outcomes = 216
- Favorable outcomes = {(4,6,6), (5,5,6), (5,6,5), (6,4,6), (6,5,5), (6,6,4)} = 6
- Probability = 6/216 = 1/36
Probability Based on Cards
Standard Deck Structure
- Total cards = 52
- 4 suits: Hearts (♥), Diamonds (♦), Clubs (♣), Spades (♠)
- Each suit has 13 cards: Ace, 2-10, Jack, Queen, King
- Face cards: Jack, Queen, King (12 total)
- Red cards: Hearts & Diamonds (26 total)
- Black cards: Clubs & Spades (26 total)
Example 1: Probability of drawing a King?
- Number of Kings = 4
- Total cards = 52
- Probability = 4/52 = 1/13
Example 2: Probability of drawing a Diamond card?
- Number of Diamonds = 13
- Probability = 13/52 = 1/4
With Replacement vs Without Replacement
With Replacement
Card is returned to deck after drawing. Probability remains constant for each draw.
Without Replacement
Card is NOT returned. Probability changes for subsequent draws.
Important Note
If question doesn\'t specify, assume WITHOUT REPLACEMENT for card problems.
Example: Two cards drawn randomly. Probability both are red?
A) With Replacement
P(second red) = 26/52 = 1/2
P(both red) = (1/2) × (1/2) = 1/4
B) Without Replacement
P(second red) = 25/51
P(both red) = (1/2) × (25/51) = 25/102
Miscellaneous Probability Problems
General Bag/Container Problems
These involve picking balls, pens, or objects from containers with different colors/types.
Example 1: Bag contains 7 red and 8 blue pens. One pen drawn randomly. Probability it\'s red?
- Total pens = 7 + 8 = 15
- Red pens = 7
- Probability = 7/15
Example 2: Container has 4 orange, 7 purple, and 9 black balls. Two balls drawn. Probability first is black, second is orange?
A) With Replacement
- Total balls = 4 + 7 + 9 = 20
- P(black) = 9/20
- P(orange) = 4/20 = 1/5
- P(black then orange) = (9/20) × (1/5) = 9/100
B) Without Replacement
- P(black) = 9/20
- After removing one black ball: total = 19, orange = 4
- P(orange) = 4/19
- P(black then orange) = (9/20) × (4/19) = 9/95
Frequently Asked Questions
What\'s the difference between \"at least\" and \"at most\" in probability?
\"At least one\" means 1 or more (use complement rule: 1 - P(none)). \"At most one\" means 0 or 1. Always carefully read these phrases.
How to remember outcomes for two dice sums?
Use the symmetric pattern: 2(1), 3(2), 4(3), 5(4), 6(5), 7(6), 8(5), 9(4), 10(3), 11(2), 12(1). Total always 36 outcomes.
When should I multiply probabilities vs add them?
Multiply for \"and\" events (both must occur). Add for \"or\" events (either can occur), but subtract overlap if events aren\'t mutually exclusive.
What\'s the complement rule in probability?
P(event) = 1 - P(event doesn\'t happen). Useful for \"at least one\" problems: P(at least one) = 1 - P(none).
How to handle probability with multiple draws?
Always check if replacement is mentioned. With replacement = independent events. Without replacement = dependent events (probability changes).
Probability Shortcuts & Tips
Coin Probability Shortcuts
Quick formulas for 2, 3, or more coin tosses without listing all outcomes.
Dice Sum Patterns
Memorize triangular patterns for quick two-dice sum calculations.
Card Deck Mnemonics
Easy ways to remember card deck structure and probabilities.
Common Probability Traps
Avoid these frequent mistakes in probability calculations.
Frequently Asked Questions
What is Probability?
Probability is an important aptitude topic used in competitive exams that tests your logical reasoning and problem-solving abilities.
Is Probability important for competitive exams?
Yes, Probability is frequently asked in SSC, Bank, CAT, TCS, and other placement exams. It's essential to master this topic for better scores.
How to prepare Probability easily?
Practice solved examples, learn formulas and shortcuts, and attempt practice questions regularly to master Probability.
What are the important formulas in Probability?
Key formulas vary by topic, but generally include basic concepts, shortcuts, and standard problem-solving approaches specific to Probability.
How many questions come from Probability?
Typically 5-10 questions come from Probability in most competitive exams, making it a high-scoring section.
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