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.

If you think you will either see one and only one falling star in the hour or not, the chance would be $0.32$ as the $0.64$ is equally spread. The other answer assumes there is a constant density of falling stars in time and you might see more than one. This shows how the assumptions change the answer.

If you think you might see one falling star in the first minute, then are guaranteed not to see one at any other time in the hour, it would depend which half hour the question is asking about.