Three students try a hard question on their own. The probabilities that they answer it correctly are \(\frac{1}{3}\), \(\frac{1}{4}\) and \(\frac{1}{5}\) respectively, independently of one another. What is the probability that the question is answered correctly by at least one of them?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\frac{3}{5}\)
“At least one answers correctly” fails only when all three get it wrong, so its probability is \(1-\) P(all wrong). For independent students, P(all wrong) is the product of the three failure probabilities.
Failure probabilities: \(\displaystyle 1-\frac{1}{3}=\frac{2}{3}\), \(\displaystyle 1-\frac{1}{4}=\frac{3}{4}\), \(\displaystyle 1-\frac{1}{5}=\frac{4}{5}\)
P(all three wrong) \(\displaystyle =\frac{2}{3}\times\frac{3}{4}\times\frac{4}{5}=\frac{2}{5}\)
P(at least one correct) \(\displaystyle =1-\frac{2}{5}=\frac{3}{5}\)