They take this approach to balance out the intake , mixture of non-selective prep-prep and very selective prep.
Its a very popular school as its next door to St Pauls girls and the girls at Bute use their pool, so its seen incorrectly as St Pauls girls junior school. (St Pauls produce the best A level results in the whole country).
Saying that, they are both Mercer Company schools , all of which produce (like their boys schools - St Pauls Boys, or Hall school Prep ) top results. As a result, very popular, especially their 7+ entry.
For the lotto, about 300+ girls for less than 15 or so places.
You can register your child at birth for Bute ballot, most register before 1st birthday ( or at least in my day). They interview parents even if you 'win' the ballot, I'm not sure what would happen if you came across as a loon to head, withdraw the place?