Probability that a pump will fail during its design life 

By multiplication rule
\[P(\#1 | \#2)= \frac{P(\# 1 \cap \# 2)}{P(\#2)}=\frac{1\%}{7\%}=\frac{1}{7}.\]
By Bayes' Theorem and the law of total probability 
\[P(\#1)=P(\#1 | \#2) P(\#2)+P(\#1 | (\# 2)^c)P((\#2)^c)\]
\[=\frac{1}{7} 7\%+7\% \times 93\%=1\%+6.51 \%=7.51\%.\]
Here $P(\#1)$ and $P(\#2)$ denotes the probability that pump 1 and 2 will fails. respectively. Similarly $P(\#2^c)$ denotes the probability that pump 2 will not fails. Also $P(\#1 | \#2)$ denotes the probability that the pump 1 will fail if pump 2 fails.