Advanced Probability | Probability Aptitude Practice | Kuriloo

Fundamental Principles

Probability Addition Theorem

For any two arbitrary events A and B, the combined probability is given by: P(A ∪ B) = P(A) + P(B) - P(A ∩ B).

Bayes' Theorem Basis

Calculated using the inverse formula structure: P(A|B) = [P(B|A) × P(A)] / P(B).

Essential Formulation Tips

If two events are mutually exclusive, their intersection P(A ∩ B) is exactly 0. This means you can simplify your addition formula to just: P(A) + P(B).

Shortcut Execution Techniques

When working on problem statements where multiple independent targets operate simultaneously, calculate the probability of each failure path first to find your answer quickly.

Contextual Inquiries (FAQs)

Q: What does a mutually exclusive event group mean?

A:

Example Breakdown: Applying the Addition Theorem

Core puzzle evaluation format common in modern technical aptitude tests.

Problem QueryThe probability that student A solves an engineering puzzle is 1/4, and the probability that student B solves it is 1/3. If both try independently, find the probability that the puzzle gets solved.

Step-By-Step Solution Path

Step 1: Identify that the puzzle is solved if A succeeds, B succeeds, or both succeed.

Step 2: Calculate independent failure paths: P(A') = 3/4, P(B') = 2/3.

Step 3: Find the probability that both fail: (3/4) × (2/3) = 6/12 = 1/2.

Step 4: Subtract the total failure path from 1: 1 - 1/2 = 1/2.

Analytical Hint: Calculate the probability that both students fail to solve the puzzle, then subtract that value from 1.

Advanced Probability Practice Set 1

10 questions evaluating Bayes' theorem patterns, independent target intersections, and set theories.

Q1. Given P(A) = 0.4, P(B) = 0.5, and P(A ∩ B) = 0.15, find the value of P(A ∪ B).

Q2. A machine has two independent components, X and Y. The failure probability of X is 0.1 and Y is 0.2. Find the probability that both components fail.

Q3. An archer hits a target 70% of the time. If she shoots twice independently, find the probability that she hits the target at least once.

Q4. Events A and B are mutually exclusive. If P(A) = 0.35 and P(B) = 0.45, find the value of P(A ∩ B).

Q5. A business manager estimates that project success depends on two independent conditions: market demand (prob 0.8) and supply chain speed (prob 0.9). Find the probability that the project succeeds.

Q6. The probability that a company truck breaks down is 0.05. Find the probability that out of two independent trucks, at least one breaks down.

Q7. If P(A') = 0.65, P(B) = 0.5, and P(A ∪ B) = 0.6, calculate the value of the intersection P(A ∩ B).

Q8. An insurance agency reports that 1% of assets suffer storm losses. A detection alarm sounds 90% of the time during an actual storm, but has a 5% false alarm rate during calm weather. Which theorem calculates the probability of a storm given that the alarm sounded?

Q9. A production line uses two independent check routines. System A catches 90% of structural errors and System B catches 80%. Find the probability that an error is caught by at least one system.

Q10. Events M and N are independent. If P(M) = 0.3 and P(N) = 0.4, find the combined value of P(M ∪ N).