Probability
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Solved Examples
Study MaterialSolved Examples – Probability
Solved examples help students understand the practical application of probability concepts and formulas in competitive examinations. These examples are designed from basic to advanced level and cover important questions frequently asked in SSC, Banking, Railway, CAT, NDA, CDS, Insurance, Defence, and various aptitude examinations.
Topics Covered in Solved Examples
- Coin Toss Probability
- Dice Problems
- Playing Card Questions
- Independent and Dependent Events
- Combination-Based Probability
- At Least One Problems
- Conditional Probability
- Advanced Probability Problems
Example 1: Probability of Getting Head
Question: A coin is tossed once. Find the probability of getting a Head.
Solution:
Sample Space:
S = {H, T}
Total outcomes = 2
Favourable outcome for Head = 1
Probability:
= 1/2
Answer: 1/2
Example 2: Probability of Getting Tail
Question: A coin is tossed once. Find the probability of getting Tail.
Solution:
Sample Space:
{H, T}
Favourable outcome for Tail = 1
Total outcomes = 2
Probability:
= 1/2
Answer: 1/2
Example 3: Probability on a Die
Question: A die is thrown once. Find the probability of getting number 4.
Solution:
Sample Space:
{1,2,3,4,5,6}
Total outcomes = 6
Favourable outcome = 1
Probability:
= 1/6
Answer: 1/6
Example 4: Probability of Even Number
Question: A die is thrown once. Find the probability of getting an even number.
Solution:
Even numbers:
{2,4,6}
Favourable outcomes = 3
Total outcomes = 6
Probability:
= 3/6
= 1/2
Answer: 1/2
Example 5: Probability of Prime Number on Die
Question: Find the probability of getting a prime number when a die is thrown.
Solution:
Prime numbers on die:
2, 3, 5
Favourable outcomes = 3
Total outcomes = 6
Probability:
= 3/6
= 1/2
Answer: 1/2
Example 6: Two Coin Tosses
Question: Two coins are tossed simultaneously. Find the probability of getting two Heads.
Solution:
Sample Space:
{HH, HT, TH, TT}
Total outcomes = 4
Favourable outcome:
HH
Probability:
= 1/4
Answer: 1/4
Example 7: Probability of At Least One Head
Question: Two coins are tossed simultaneously. Find the probability of getting at least one Head.
Solution:
Using complementary probability:
P(At least one Head)
= 1 − P(No Head)
No Head means:
TT
Probability of TT:
= 1/4
Therefore:
= 1 − 1/4
= 3/4
Answer: 3/4
Example 8: Two Dice Problem
Question: Two dice are thrown simultaneously. Find the probability of getting sum 7.
Solution:
Total outcomes:
= 6 × 6
= 36
Favourable outcomes:
(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)
Total favourable outcomes = 6
Probability:
= 6/36
= 1/6
Answer: 1/6
Example 9: Probability of Doublet
Question: Two dice are thrown simultaneously. Find the probability of getting a doublet.
Solution:
Doublets:
(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)
Favourable outcomes = 6
Total outcomes = 36
Probability:
= 6/36
= 1/6
Answer: 1/6
Example 10: Playing Card Probability
Question: One card is drawn from a pack of 52 cards. Find the probability of getting an Ace.
Solution:
Total cards = 52
Total Aces = 4
Probability:
= 4/52
= 1/13
Answer: 1/13
Example 11: Probability of Red Card
Question: A card is drawn from a deck of 52 cards. Find the probability of getting a red card.
Solution:
Red cards:
= 26
Total cards:
= 52
Probability:
= 26/52
= 1/2
Answer: 1/2
Example 12: Probability of Face Card
Question: Find the probability of drawing a face card from a deck of cards.
Solution:
Face cards:
King, Queen, Jack
Total face cards:
= 12
Probability:
= 12/52
= 3/13
Answer: 3/13
Example 13: Independent Events
Question: A coin is tossed and a die is rolled simultaneously. Find the probability of getting Head and number 5.
Solution:
Probability of Head:
= 1/2
Probability of getting 5:
= 1/6
Using multiplication rule:
= 1/2 × 1/6
= 1/12
Answer: 1/12
Example 14: Probability Without Replacement
Question: Two cards are drawn one after another without replacement from a pack of cards. Find the probability that both are Kings.
Solution:
Probability of first King:
= 4/52
= 1/13
Probability of second King:
= 3/51
= 1/17
Required Probability:
= 1/13 × 1/17
= 1/221
Answer: 1/221
Example 15: Complementary Probability
Question: Find the probability that at least one Tail appears when three coins are tossed.
Solution:
Using complementary probability:
P(At least one Tail)
= 1 − P(No Tail)
No Tail means:
HHH
Total outcomes:
= 23
= 8
Probability of HHH:
= 1/8
Required Probability:
= 1 − 1/8
= 7/8
Answer: 7/8
Example 16: Combination-Based Probability
Question: Three cards are drawn from a deck of cards. Find the probability that all are Aces.
Solution:
Total ways to draw 3 cards:
= 52C3
Ways to draw 3 Aces:
= 4C3
Probability:
= 4C3 / 52C3
= 4 / 22100
= 1 / 5525
Answer: 1/5525
Example 17: Probability of Sum Greater Than 9
Question: Two dice are thrown simultaneously. Find the probability that the sum is greater than 9.
Solution:
Possible sums greater than 9:
10, 11, 12
Favourable outcomes:
10 → (4,6), (5,5), (6,4)
11 → (5,6), (6,5)
12 → (6,6)
Total favourable outcomes = 6
Total outcomes = 36
Probability:
= 6/36
= 1/6
Answer: 1/6
Important Exam Tips
- Always calculate total outcomes first.
- Use complementary probability for “at least” problems.
- Memorize card and dice concepts.
- Practice independent and dependent event problems regularly.
- Use combinations carefully.
- Read conditions properly before solving.
- Improve mental calculation speed.
Common Mistakes to Avoid
- Using wrong sample space.
- Ignoring replacement conditions.
- Confusing independent and dependent events.
- Using incorrect favourable outcomes.
- Calculation mistakes in combinations.
Practicing solved examples regularly improves conceptual clarity, logical thinking, and calculation speed in solving Probability aptitude questions in competitive examinations.