Three archers shoot one arrow each at a target. The probabilities that they hit the target are \(\frac{1}{2}\), \(\frac{1}{3}\) and \(\frac{1}{4}\) respectively, independently of one another. What is the probability that the target is hit at least once?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\frac{3}{4}\)
“Hit at least once” fails only when all three archers miss, so its probability is \(1-\) P(all miss). For independent archers, P(all miss) is the product of the three miss probabilities.
Miss probabilities: \(\displaystyle 1-\frac{1}{2}=\frac{1}{2}\), \(\displaystyle 1-\frac{1}{3}=\frac{2}{3}\), \(\displaystyle 1-\frac{1}{4}=\frac{3}{4}\)
P(all three miss) \(\displaystyle =\frac{1}{2}\times\frac{2}{3}\times\frac{3}{4}=\frac{1}{4}\)
P(target hit at least once) \(\displaystyle =1-\frac{1}{4}=\frac{3}{4}\)