Question

If A,B and C are three independent events such that P(A) = P(B) = P(C) = P then P(at least two of A, B, C occur) =

• A. P³ - 3P
• B. 3P - 2P²
• C. 3P² - 2P³
• D. 3P²

Answer 1

The Correct Option is C

We are given that the events A, B, and C are independent, and \( P(A) = P(B) = P(C) = p \). The probability that at least two of the events A, B, and C occur is equivalent to the probability that either exactly two events occur or all three events occur. 1. **Probability of exactly two events occurring: The probability of exactly two events occurring can be calculated using the binomial probability formula for combinations. There are three possible combinations of two events happening: \[ P(\text{exactly two events occur}) = 3 \cdot p^2 \cdot (1 - p) \] 2. **Probability of all three events occurring: Since the events are independent, the probability of all three events occurring is: \[ P(\text{all three events occur}) = p^3 \] 3. Total probability: The total probability that at least two events occur is the sum of the probabilities of exactly two events occurring and all three events occurring: \[ P(\text{at least two events occur}) = 3 \cdot p^2 \cdot (1 - p) + p^3 \] Simplifying: \[ P(\text{at least two events occur}) = 3p^2 - 3p^3 + p^3 = 3p^2 - 2p^3 \] Thus, the correct answer is (C): \( 3p^2 - 2p^3 \).

Answer 2

Given: Events A, B, and C are independent, with \(P(A) = P(B) = P(C) = P\) 

We need: \(P(\text{at least two of A, B, C occur})\)

This includes the cases when exactly two occur + all three occur.

Step 1: Probability that all three occur:

\(P(A \cap B \cap C) = P^3\)

Step 2: Probability that exactly two occur:

• \(A \cap B \cap C' = P \cdot P \cdot (1 - P) = P^2(1 - P)\)
• \(A \cap C \cap B' = P^2(1 - P)\)
• \(B \cap C \cap A' = P^2(1 - P)\)

So total probability for exactly two = \(3P^2(1 - P)\)

Step 3: Add both:

\(P(\text{at least two occur}) = 3P^2(1 - P) + P^3\)

\(= 3P^2 - 3P^3 + P^3 = 3P^2 - 2P^3\)

Answer: \(3P^2 - 2P^3\)