Consider Lottery A which has a probability $0.8$ of winning a prize and Lottery B where the probability of winning a prize is $0.9$. If I buy one ticket for Lottery A and one ticket for Lottery B, what is the probability that I will win at least one prize?
I came up with this toy problem but what I am ultimately trying to figure out are the tools used to solve a generalization to, let's say, $n$ lotteries each with a probability $p_n$ of winning a prize.
You can consider the probability of not winning any prize. In this case it would be $0.1 \cdot 0.2 = 0.02$, therefore the probability that you win at least one prize is $1-0.02 = 0.98$