In an experiment on mosquito insecticide, it is found that 80% are killed during the initial application, but those that survive develop a resistance. The percentage of the survivors killed in any later application is half of that of the immediately preceding application; i.e., 40% of the survivors of the first application would succumb to the second application, while only 20% of the survivors of two applications would be killed by a third application, etc. Consider each application independent of any other application.
1. Find the probability that a mosquito will survive 5 applications.
2. Find the probability that a mosquito will survive 5 applications, given that it has survived the first 2 applications.