Lilly and Rose order two electronic components but only one of the components they receive is genuine, and the other is duplicate. Although the components look exactly same , the duplicate one is not as reliable. The lifetime (in months) of the genuine part is modeled as a geometric random variable with probability 0.01 of failure in a given month, while the lifetime (in months) of the duplicate part is modeled independently as a geometric random variable with probability 0.02 of failure in a given month. Lilly randomly chooses one component to install, and Rose installs the other. Lilly 's part ends up failing in month 40, while Rose's part ends up failing in month 60. Given this information, what is the probability that Lilly picked the genuine part? What is the probability that Rose picked the genuine part?

(Could you please explain the steps to understand?)