Question

Ajit, Ravi and Hari were trying to hit a target. Ajit hits the target 5 times in 8 attempts, Ravi hits it 3 times in 5 attempts, and Hari hits it 2 times in 4 attempts. What is the probability that the target is hit by at least 2 persons?

• A. \(\frac{49}{80}\)
• B. \(\frac{24}{80}\)
• C. \(\frac{45}{80}\)
• D. \(\frac{25}{80}\)

Hint:

Use the complement rule: P(at least 2) = 1 minus P(0 hits) minus P(exactly 1 hit).

Answer 1

The Correct Option is A

Step 1: List the hit and miss chances.
Ajit: hit $\frac{5}{8}$, miss $\frac{3}{8}$. Ravi: hit $\frac{3}{5}$, miss $\frac{2}{5}$. Hari: hit $\frac{1}{2}$, miss $\frac{1}{2}$.

Step 2: Add up the exactly-2-hits cases directly.
Ajit and Ravi hit, Hari misses: $\frac{5}{8} \times \frac{3}{5} \times \frac{1}{2} = \frac{15}{80}$. Ajit and Hari hit, Ravi misses: $\frac{5}{8} \times \frac{2}{5} \times \frac{1}{2} = \frac{10}{80}$. Ravi and Hari hit, Ajit misses: $\frac{3}{8} \times \frac{3}{5} \times \frac{1}{2} = \frac{9}{80}$. Sum of exactly 2 hits = $\frac{15+10+9}{80} = \frac{34}{80}$.

Step 3: Add the case where all three hit.
All hit: $\frac{5}{8} \times \frac{3}{5} \times \frac{1}{2} = \frac{15}{80}$.

Step 4: Add exactly-2 and all-3 cases together.
"At least 2" means exactly 2 or exactly 3, so add these: $\frac{34}{80} + \frac{15}{80} = \frac{49}{80}$. This matches the complement method, confirming the answer.

Final Answer:
The probability is $\frac{49}{80}$. \[ \boxed{\frac{49}{80}} \]