I doubt there is any easy formula such as "hit rate among applicants is 100% among applicants who have also applied to Oxbridge".
The LSE website will tell you how many applicants and how many places. Many courses are very competitive.
The important thing obviously with any competitive University that does not interview is to get the Personal Statement right. It will be scored. There will be guidance. Follow it closely to make sure you pick up every point you can. Go through the website carefully and give as many good reasons as you can for why you want to do the course. Evidence any specific interest you have which backs up your reasons.
Its not the University that will be ruthless. The competition may be.
One tip with LSE, is that they have a great series of public lectures, worth going to if you can get there. A chance to get a feel of the place and to develop/provide evidence of any interest.
I have noticed that bojorojo has consistently posted slightly negative comments about LSE as if one of her DC has had a poor experience perhaps during the application process. It is hard to get a place, but for a child that wants to be in the centre of a world city and likes the course, it is fantastic, great academically, and truely international.
(DS goes there and indeed many of his friends are Chinese, but often born in places like Manchester or Hounslow!)