Probability Solved Examples (Class 12 Mathematics)
Probability numericals cover basic probability, conditional probability, and probability distributions. These problems develop statistical reasoning essent
TL;DR: Probability numericals cover basic probability, conditional probability, and probability distributions. These problems develop statistical reasoning e…
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Probability numericals cover basic probability, conditional probability, and probability distributions. These problems develop statistical reasoning essent
Probability — Solved Numerical Examples (Step by Step)
Example 1: A die is rolled. Find the probability of getting a number greater than 4.
Solution: Total outcomes when rolling a die = 6. Favorable outcomes (5 or 6) = 2. Probability = 2/6 = 1/3 ≈ 0.333.
Example 2: A card is drawn from a standard deck of 52 cards. Find the probability it is a heart.
Solution: Total cards = 52. Number of hearts = 13. Probability = 13/52 = 1/4 = 0.25.
Example 3: Two dice are rolled. Find the probability that the sum is 7.
Solution: Total outcomes = 36. Favorable outcomes: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6. Probability = 6/36 = 1/6 ≈ 0.167.
Example 4: A bag has 3 red balls and 5 blue balls. If 2 balls are drawn without replacement, find probability both are red.
Solution: Total balls = 8. Probability first is red = 3/8. After drawing one red, 2 red remain and 7 total remain. Probability second is red = 2/7. P(both red) = 3/8 × 2/7 = 6/56 = 3/28 ≈ 0.107.
Example 5: Given P(A) = 0.4 and P(B) = 0.3 with A and B independent, find P(A ∩ B).
Solution: For independent events: P(A ∩ B) = P(A) × P(B) = 0.4 × 0.3 = 0.12.
Example 6: In a class of 100 students, 60 play cricket and 50 play football. 20 play both. Find probability a randomly selected student plays at least one sport.
Solution: Using inclusion-exclusion: P(C ∪ F) = P(C) + P(F) - P(C ∩ F) = 60/100 + 50/100 - 20/100 = 90/100 = 0.9.
Tips
- Basic probability: P(A) = Favorable outcomes / Total outcomes.
- P(A and B) = P(A) × P(B) for independent events.
- P(A or B) = P(A) + P(B) - P(A and B) (inclusion-exclusion principle).
Frequently Asked Questions
What is the difference between dependent and independent events?
Independent events: outcome of one does not affect the other. Dependent events: outcome of one changes probability of the other. Example: rolling dice (independent) vs. drawing cards without replacement (dependent).
What is conditional probability?
Conditional probability P(A|B) is the probability of A given that B has occurred. It is calculated as P(A|B) = P(A ∩ B) / P(B).
More Mathematics Solved Examples
- Quadratic Equations
- Trigonometry
- Arithmetic Progressions
- Surface Areas and Volumes
- Coordinate Geometry
- Probability
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