Probability of space junk hitting the southern hemisphere About $20\%$ of the southern hemisphere is land. South Africa takes
up about $5\%$ of the the land surface of the southern hemisphere. A
piece of space junk is falling to earth such that it could hit anywhere
randomly. Calculate the probability that it will hit (a) the southern
hemisphere, (b) the land in the southern hemisphere, (c) the sea in the
southern hemisphere, (d) South Africa.
 A: This is quite a simple problem. Without giving much away, I will provide some insight.
Consider $\Omega$ as the planet Earth. This question assumes that $P(\Omega)=1$, the probability of the space junk landing on Earth.
We can partition $\Omega$ into the southern hemisphere $S$ and the northern hemisphere $S^C$, where $S^C$ is the portion of Earth that is not the Southern hemisphere.
a) Since these two portions are equal, hence the word hemisphere, the probability of landing on any one of these hemispheres is precisely $\dfrac{1}{2}$.
b) Let $L$ be the portion of Earth that is land. We have the information that $20\%$ of the southern hemisphere is land. Thus, we want to find $P(S\cap L)$, the probability of landing on land in the southern hemisphere, which is a straightforward problem.
c) Reasonably, we can assume wherever is not land is the "sea" or other bodies of water. So, we would like to find $P(S\cap L^C)$, which in its on right is also a simple calculation.
d) Let $N$ be the nation of South Africa. It is obvious that $N\in S$ since South Africa is located in the southern hemisphere. Thus, we would like to find $P(S\cap L\cap N)=P(N)$ since $N\in S$ and $N\in L$.
A: (a) The probability that the space junk hits the southern hemisphere is clearly $50\%$ since the southern hemisphere takes up half of the globe.  
(b) The land in the southern hemisphere is $20\%$ of the total area of the southern hemisphere. To find how big of a percentage this is compared to the whole globe, we can take $20\%/ 2 =10\%$. Alternatively you can think of it as the probability of the space junk landing in the southern hemisphere AND on the land there. This is $50\%\times 20\%=10\%$.  
If you understand (b) then (c) and (d) are similar. Try it. 
