Three children take part in a race at their school. The probabilities that they reach the finishing point are \(\frac{2}{5}\), \(\frac{1}{6}\) and \(\frac{3}{7}\) respectively, independently of one another. What is the probability that at least one of them finishes the race?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\frac{5}{7}\)
“At least one finishes” fails only when none of them finishes. So find P(none finishes) as a product and take it away from 1.
P(first does not finish) \(\displaystyle =1-\frac{2}{5}=\frac{3}{5}\)
P(second does not finish) \(\displaystyle =1-\frac{1}{6}=\frac{5}{6}\)
P(third does not finish) \(\displaystyle =1-\frac{3}{7}=\frac{4}{7}\)
P(none finishes) \(\displaystyle =\frac{3}{5}\times\frac{5}{6}\times\frac{4}{7}=\frac{60}{210}=\frac{2}{7}\)
P(at least one finishes) \(\displaystyle =1-\frac{2}{7}=\frac{5}{7}\)