Astring of lights contains 3 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 2-year period. if the lights fail independently of each other, what is the probability that a string of lights will remain bright for 2 years?

Assume the three lights' failure rates are independent of each other (e.g. no surges in current to cause them to all fail).
Since if any light of the three fails, the whole string goes dark.
So the "surviving" probability of each light is 1-0.1=0.9
For all three lights to survive 2 years, the probability can be found by the multiplication rule (applies when each probability is independent of each other for a multi-step experiment)
P(3)=(0.9*0.9*0.9)=0.729

If you make it a little more clear i can answer it