A rocket's engine has 3 stages. They have a 0.99, 0.97 and 0.98 probability of success respectively. Find the reliability of the rocket.
I'm new to the topic, so I think I might be missing something, but isn't the solution: $0.99 \cdot 0.97 \cdot 0.98$?
Assuming the stages are independent, then yes, that's all there is to it.
When you have several independent things that all need to happen (simultaneously, or in succession, doesn't matter), the probability that they all happen is in general equal to the product of each of them happening. In this case you need each of the three stages to succeed.
If they are dependent, then we need more information about how they depend on wone another. In particular, is it relatively common that they all fail, or is it more common that only one part fails?
However, if these numbers are given as:
then you still get probability $0.99\cdot 0.97\cdot 0.98$ for the entire launch succeeding, regardless of dependence. The three events being independent is just a special case of this.