The most universally relevant subject to study would be the probability branch of mathematics and in this its highly applicable for the pure numbers side to a game even minus a person you can 'autopilot' out the trajectory of most games. Sociology or psychology I think would be more relevant there as to what motivates people in terms of playing vs knowing it is a business with a cost to it. You could theorize the flight fight response and how this raises adrenaline levels and could be linked to gambling. Like alot of high energy sports there is a large amount of risk and competetive element in there that we as humans are motivated to take part in from our heritage as hunter gatherers.
Economics I find only relevant via gambling as an instrument of taxation in an economy. Law because gambling is often restricted is relevant from that sideline.
Maybe, it would be more appropriate if his thesis dealt with sociology than the others because it would also relate to what would happen to humans if they gambled. But when it comes to math about probability theory that a gambler can get, I don't think many people will try to dig up the information.
Maybe if it is related to socio-economic or sociological issues, there is a lot to discuss and maybe it will be interesting material to read by many people. If it was economics, it might also be interesting but in my opinion, that would be too broad a discussion.
Gambling can be interpreted from so many perspectives that a thesis would probably only be able to focus on one aspect. If maths and statistics are going to be the focus, I can already tell you that it would require a very original perspective to create something new. Probabilities, statistics and other are well studied fields because these influence many industries already.
As for the final result, the thesis will indeed focus on one aspect only but before discussing it, it can relate to several things closely related to gambling. The thesis can have mathematics and statistics in the discussion of the thesis. But still, I think other topics will be more interesting, especially if they are related to the economy or socio-economics of the gambling players. Later, at the end of the discussion, it will be written what should be done and what should be avoided by the community so that they are not addicted to gambling.