Question

Ajit, Ravi and Hari were trying to hit a target. If 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. \(\frac {25}{80}\)

Answer 1

The Correct Option is A

To solve the problem, we need to find the probability that at least 2 out of Ajit, Ravi, and Hari hit the target. Let's start by calculating the individual probabilities of success and failure for each person: 

• Ajit's probability of hitting the target, \(P(A)\), is \(\frac{5}{8}\), and missing, \(P(\bar{A})\), is \(1 - \frac{5}{8} = \frac{3}{8}\).
• Ravi's probability of hitting the target, \(P(R)\), is \(\frac{3}{5}\), and missing, \(P(\bar{R})\), is \(1 - \frac{3}{5} = \frac{2}{5}\).
• Hari's probability of hitting the target, \(P(H)\), is \(\frac{2}{4} = \frac{1}{2}\), and missing, \(P(\bar{H})\), is \(1 - \frac{1}{2} = \frac{1}{2}\).

Next, compute the probability that the target is hit by at least 2 persons. We will use complementary probability to simplify the calculations:

First find the probability that the target is hit by fewer than 2 persons, i.e., none or only one hits the target:

• None hit:\(P(\bar{A})P(\bar{R})P(\bar{H})=\frac{3}{8}\times\frac{2}{5}\times\frac{1}{2}=\frac{3}{40}\)
• Only Ajit hits:\(P(A)P(\bar{R})P(\bar{H})=\frac{5}{8}\times\frac{2}{5}\times\frac{1}{2}=\frac{1}{8}\)
• Only Ravi hits:\(P(\bar{A})P(R)P(\bar{H})=\frac{3}{8}\times\frac{3}{5}\times\frac{1}{2}=\frac{9}{80}\)
• Only Hari hits:\(P(\bar{A})P(\bar{R})P(H)=\frac{3}{8}\times\frac{2}{5}\times\frac{1}{2}=\frac{3}{40}\)

Now add these probabilities:

\(P(\text{fewer than 2 hit})=\frac{3}{40}+\frac{1}{8}+\frac{9}{80}+\frac{3}{40}=\frac{3}{40}+\frac{10}{80}+\frac{9}{80}+\frac{6}{80}=\frac{28}{80}\)

To find the probability that at least 2 hit the target:

\(P(\text{at least 2 hit})=1-P(\text{fewer than 2 hit})=1-\frac{28}{80}=\frac{52}{80}=\frac{49}{80}\)

Thus, the probability that the target is hit by at least 2 persons is \(\frac{49}{80}\).