Raja, Karan, and Ashwini each fire one shot at a target. Their hit probabilities are: Raja = 3/6, Karan = 2/6, Ashwini = 4/4. What is the probability that at least two shots hit the target?

Introduction / Context:
We have three independent Bernoulli trials with different success probabilities: p_R = 1/2, p_K = 1/3, p_A = 1. We want P(at least two hits). Because Ashwini always hits, the total hits are 1 (from Ashwini) plus the number of hits from Raja and Karan. Hence, “at least two” reduces to “Raja or Karan (or both) hits”.

Given Data / Assumptions:

• p_R = 3/6 = 1/2.
• p_K = 2/6 = 1/3.
• p_A = 1.
• Independence of shooters.

Concept / Approach:
Compute P(exactly 2 hits) and P(exactly 3 hits) and sum them. Exactly 2 occurs when Ashwini hits and exactly one of {Raja, Karan} hits. Exactly 3 occurs when all three hit.

Step-by-Step Solution:

Verification / Alternative check:
Shortcut: Because p_A = 1, at least two hits ⇔ (R or K hits). P(R ∪ K) = 1/2 + 1/3 − (1/2)(1/3) = 5/6 − 1/6 = 2/3.

Why Other Options Are Wrong:
1/2 ignores triple-hit scenarios; 5/6 overcounts; 1/3 is too small.

Common Pitfalls:
Forgetting independence or misinterpreting “at least two” as “exactly two”.

Final Answer:
2/3