Calculating the probability of seeing a shooting star within half an hour if we know it for one hour The probability to see a falling star in the sky over the course of one hour is 0.64. What is the probability to see it over the course of half an hour?

An hour is two half hours. If the probability in each half hour is $p$, the probability not to see a falling star in the entire hour is $(1-p)^2=1-0.64=0.36$, so $1-p=0.6$ and $p=0.4$. Note that $0.64$ was chosen such that the square root could be easily drawn with or without taking complements, so you wouldn't have noticed if you hadn't taken complements.