Answers have been useful. You have learnt it.
I am just sharing my learning here, just because the discussion help us solidify our understanding. Usually we talk about coins, dies in terms of trying to either understand or explain probabilistic events. If I don't use certain concepts for a while, I forget them. Then I hit another subject, module say biology where Mendelian inheritance (genetics) actually incorporate probability, specifically independent events and mutually exclusive events.
It was quite interesting and let me revise my own concept gaps. I hope this will let you think from another application point of view.
Product rule applies to independent events. You can be tall regardless of your skin colour. You can be fair regardless of your height. So you can be tall and fair at the same time, (no offence to anyone, just taking the dominant traits here) These are two independent events, that can come about together.
Another way, you like to swim, your brother/sister likes to play rugby. Both of you could be Olympians, racing in your competitions at the same time and taking two medals away together. You can't swim in a rugby ground, so you will not influence his medal in anyway, so is he.
You like to play the piano, your friend likes to play the violin. Both of you can play together and conduct a concert or an orchestra (if you may). You don't influence each other in anyway.
Thus, swim AND rugby taking place
= p(swim) * p(rugby)
Sum rule applies to mutually exclusive events. Imagine, only one swimmer will be chosen from the country to represent in Olympics for 100m freestyle. Lets say you are from Australia, and coaches must select one person for the team. One person, either you or Kyle Chalmers.
In this case if you are selected, Kyle can't and vice versa. Both of you can't swim at the same time.
Thus probability of you OR Kyle swimming in Olympics
= p(you) + p(Kyle)
Coming back to the genetics simple example for mutual exclusiveness, you can either be tall or short, fair or dark.
Maths is life, explains physics, chemistry, biology, and what not. So Good luck!