My assumptions: (1) It's graded on a simple "percentage right" basis, so 24/40 questions necessary right to pass, (2) 4 options for each question

Her score on an individual test is a random variable X following a binomial distribution with 40 trials and chance of 0.25 for each trial. Her chance of passing by guessing randomly, P(X >= 24), is an infinitesimally small 2.826E-6.

The probability that she fails all of the 960 tests, assuming independence of tests, is (1-p)^960. So the probability that she will pass at least one test is:

  1 - (1 - p)^960

However, under the above assumptions, she could practically guarantee her success by combining this guessing strategy with a simple test-taking strategy like eliminating 1 or 2 obviously wrong answers per question, or just remembering the answers to several of the same questions that are probably being recycled from test to test.

I assumed this is how she did it, but assuming all questions are recycled, 4 options in 40 questions, perfect memory, no knowledge, and a naïve strategy, she should have passed in just 29 tries (1.75 retries per question, 10 initially right, 24 needed). Mastermind isn't that hard a game. So clearly, some assumptions are wrong.