Question

Four persons can hit a target correctly with probabilities $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}$ and $\frac{1}{5}$ respectively. If all hit at the target independently, then the probability that the target would be hit, is:

• A. $1/120$
• B. $4/5$
• C. $1/5$
• D. $2/3$

Hint:

$P(\text{At least one}) = 1 - P(\text{None})$. This complement rule is much faster than calculating all hit combinations.

Answer 1

The Correct Option is B

Step 1: Concept
The probability that the target is hit is the complement of the probability that no one hits the target.

Step 2: Meaning
Calculate the probability of failure for each person: $P(A') = 1 - 1/2 = 1/2$, $P(B') = 1 - 1/3 = 2/3$, $P(C') = 1 - 1/4 = 3/4$, and $P(D') = 1 - 1/5 = 4/5$.

Step 3: Analysis
Since the events are independent, the probability that none hit the target is the product of their individual failure probabilities: $P(\text{None}) = (1/2) \cdot (2/3) \cdot (3/4) \cdot (4/5) = 1/5$.

Step 4: Conclusion
The probability that the target is hit is $1 - P(\text{None}) = 1 - 1/5 = 4/5$. Final Answer: (B)