I have a probability question that is part of a Master's programme. The scenario is:
The probability that a factoryβs sprinkler system will fail is 0.2; the probability that its alarm system will fail is 0.1; the probability that both will fail simultaneously is 0.04. What is the probability that at least one will work?
My thinking is to calculate P(Sprinkler system not failing) + P(Alarm system not failing) + P(Not both failing simultaneously) = 0.8 + 0.1 + 0.96, but that gives a probability greater than 1 which is not possible.
Many thanks in advance.
Best Answer
the probability that at least one system will work is the complement of the probability that both fail simultaneously, thus the result is $1-0.04=0.96$
you can see the solution using Venn's diagrams