Exams

The question of what luck is, and whether or not it exists, is a challenging metaphysical question. 

Scholars who take a strict reading of Islamic theology will generally reject the idea of "luck", if one defines "luck" as something amoral (that is, acting outside a moral compass) which exists independent of the divine involvement in the universe. This is especially if it seems to go against a sense of divine justice. "Luck" is sometimes seen as belonging to pre-Islamic and/or non-Abrahamic belief systems. 

That is to say, things that are seen as "lucky" tend to be seen as Allah's will or something occurring on the level of metaphysical cause and effect (for instance, I help an orphan today, and am saved from a car crash tomorrow). This is apart from hard work and smart work; for instance, some people are "lucky" in their exams simply because they study smart.

Instead, they will focus on concepts like baraka (things, people, places, etc, being blessed by Allah), divine destiny, divine mercy, divine punishment, and so forth. So, for instance, one might increase "luck" by doing good deeds or giving charity. (That is, it is not mere luck but rather cause and effect.) 

Certain things which might be considered "luck" in some cultures are also ascribed to divine decree; for instance, Allah decrees sustenance for a human being. So, receiving sustenance is ultimately through the decree of Allah and the acts of angels in obedience to Allah in delivering it, even if we also have to put forth efforts. Or, someone may receive a scholarship to Harvard and be considered "lucky", but someone else may not receive that scholarship because it is not their destiny and they are meant to do something else in life, so it is not really about "luck". 

On the other hand, it is not impossible to find Muslims who do hold that luck exists as a metaphysical thing, and that some people seem to have more of it. After all, Muslims have held a great many views over the past 1000+ years. 

In any case, most people are not ruminating over these philosophical questions when wishing someone good luck on something, and it's fine to say.  

Absolutely; however, one also still has to study. Just writing "bismillah al-rahman al-rahim" on top of an exam does not guarantee a good score. However you can ask Allah for help with studying and help on getting good marks, and then do your best. (Which also may include asking for help from other people if you are having difficulty with the material alone.)

True, rarely, there are miracles (for instance, there may be a case where someone is desperate to pass an exam and coincidentally studies the exact topic that is on the exam, and this can be ascribed to divine assistance at a critical point in their life). But in general the divine assistance should be coupled with serious efforts.

I am saying that because it is not unusual for some students who get low grades on an example to approach prayer as a last resort, and then wonder why they are getting low marks when they write "786" on their exam and all. Of course it is always good to begin everything with bismillah, it just isn't a substitute.

But yes it is always good to ask for Allah's help.

There is a nice story about the late 'Allamah Tabataba'i who wrote the famous Tafsir al-Mizan. It is hard to believe, but he says that in his youth he had a very difficult time with school and was written off by the teacher as being a poor student. So he went out of the city and implored Allah to give him the capability to study and learn. And as we see, he became one of the most famous scholars in history. This is an example we can all follow if what we are studying is important to us!

Also remember that sometimes if studying a subject doesn't work out, sometimes there is an element of fate, that is, maybe it isn't something we are meant to do as part of our life path. (For instance, if a person doesn't get into medical school) But of course if it is important to you, you should still try your best. God knows best.