Conditional Probability – Solved Problems
Conditional probability deals with finding the probability of an event when another related event has already occurred. It is a crucial topic in statistics, data science, competitive exams, and real-life decision-making.
Problem 1: Basic Conditional Probability
A card is drawn from a standard deck of 52 cards. Find the probability that the card is an Ace, given that it is a red card.
Solution:
Red cards = 26 Red Aces = 2
Answer: Probability = 1/13
Problem 2: Dice-Based Conditional Probability
Two dice are thrown. Find the probability that the sum is 10, given that the first die shows a 4.
Solution:
If the first die is 4, possible outcomes are: (4,1), (4,2), (4,3), (4,4), (4,5), (4,6)
Sum = 10 occurs when (4,6)
Answer: Probability = 1/6
Problem 3: Students Passing Exams
In a class, 60% students pass Mathematics, 50% pass Science, and 30% pass both. Find the probability that a student passes Mathematics given that the student passed Science.
Solution:
P(M | S) = 0.30 / 0.50 = 0.6
Answer: Probability = 0.6
Problem 4: Drawing Balls from a Bag
A bag contains 5 red and 7 blue balls. One ball is drawn at random. Find the probability that it is blue, given that it is not red.
Solution:
Not red = blue only Total blue balls = 7
Answer: Probability = 1
Problem 5: Selecting a Student
Among 40 students, 25 play cricket, 20 play football, and 10 play both. Find the probability that a randomly selected student plays cricket, given that the student plays football.
Solution:
P(C | F) = 10 / 20 = 1 / 2
Answer: Probability = 1/2
Problem 6: Defective Items
A box contains 12 bulbs, out of which 3 are defective. One bulb is selected at random. Find the probability that it is defective, given that it is not good.
Solution:
Not good bulbs = defective bulbs = 3
Answer: Probability = 1
Problem 7: Real-Life Selection Problem
In a group of 100 people, 60 drink tea, 40 drink coffee, and 20 drink both. If a person is selected at random and is known to drink tea, find the probability that the person also drinks coffee.
Solution:
P = 20 / 60 = 1 / 3
Answer: Probability = 1/3
Conclusion: Conditional probability helps in understanding dependent events and real-world uncertainty. It forms the foundation for Bayes’ theorem, statistics, and data-driven decision-making.