State school vs private school intake

Hello, I am hoping to apply to Cambridge for maths, and was initially thinking Trinity, but my teacher warned me that, because it typically attracts more private school kids, there is a risk that because most of them will be getting a lot more resources and help from their school, I will be reducing my chances of getting in simply because of how competitive the applicants are. I was wondering if there was any way I could figure out which colleges tend to get more state schools offers. For example, I was looking at Caius, Peterhouse, Robinson or Trinity Hall (My current favourite is T. Hall), but I was wondering if these ones would have similar problems, and also if there is anyway where I can see the number of state school vs private school applicants for any given college. I know pooling is there to help with that, but I would like to make an informed decision, as I would ultimately prefer to make the choice on my own, of course if I were pooled, that is fine, but I would like to pick one I am actually likely to get into.