The first thing to do when faced with a question such as this, focusing on definitions and word origins, is to review the various definitions of the root word and decide which are relevant. I have used the Cambridge Dictionary (link below).

The most common definition of "lot" is "a large amount," but this does not appear to have anything to do with the word "lottery." The most promising definitions are "the quality of someone's life and the experiences they have" and, even better, the definition of the phrase "draw lots": "to make a decision by choosing from a set of objects such as pieces of paper or sticks that are all the same except for one."

It is clear that a lottery involves a process that is essentially the same as drawing lots. One can then use the definition of "lot" which refers to quality and experience of life to craft a definition of the word "lottery" using the root word "lot."