Question

A string of lights contains three lights. the lights are wired in series, so that if any light fails the whole string will go dark. each light has probability .10 of failing during a two-year period. if the lights fail independently of each other, what is the probability that a string of lights will remain bright for two years

Answer 1

Let $p_i=0.1$ be a  probability that i-th light fails, where i=1,2,3. Then, $q_i=1-p_i=1-0.1=0.9$ is the probability that i-th light remain bright.
If the lights fail independently of each other,  the probability that a string of lights will remain bright for two years is:


$q_1q_2q_3=(0.9)^3=0.729$.