It is Hume's opinion that the principle "whatever begins to exist must have a cause of existence" is not obvious and maybe even false.
He considers and rejects the following argument in favor of the principle:
All the points of time and place, say some philosophers, in which we can suppose any object to begin to exist, are in themselves equal; and unless there be some cause, which is peculiar to one time and one place, and which by that means determines and fixes the existence, it must remain in eternal suspense; and the object can never begin to be, for want of something to fix its beginning.
But I ask; Is there any more difficulty in supposing the time and place to be fixed without a cause, than to suppose the existence to be determined in that manner? The first question that occurs on this subject is always, whether the object shall exist or not? The next, when and where it shall begin to exist? If the removal of a cause be intuitively absurd in the one case, it must be so in the other: And if that absurdity be not clear without proof in one case, it will equally require one in the other. (A Treatise on Human Nature, 1.3.3, 4)
It's not just the time and place that must be determined but also the essence of the object that will begin to exist. We may call the event random, but at some point, even a second before it occurs, there must be determination. The roulette will stop spinning and reveal definite values. Then we should be able to predict that, for example, a chair precisely described will pop into existence tomorrow at 11 am and right in front of me.
Hume argues that we can imagine this to occur without any cause. The question, however, is what or who do we consult to make even this limited prediction? What do we inspect and study in order to find all this information out? Now while it is tempting to answer that it's precisely the cause of the chair, let's slow down a bit.
Suppose it's written in some database cleverly etched onto helium atoms to which we have managed to gain access that the chair will appear thusly. Why believe this information? Probably because similar entries have yielded correct predictions. Now we have 2 events appearing close to each other: a new row is added to a certain table in the database and then the thing specified, at the prescribed time and place (in that row's fields), appears out of nowhere. The connection between the database and real-world events cannot be doubted. There is no coincidence; the former anticipates and predicts the latter. This is now a law of nature, discoverable by experiment and reason. Now if the database itself is not the cause of the events recorded in it, some definite X must connect the two. Some X must one way or another force the universe to obey the instructions written in the table. X must read each entry in the table and act accordingly.
This X is what all men call the cause(s) of the chair in front of me at 11 am.
So then, yes, everything that begins to exist has a cause.