In trying to answer the questions of "to how many schools should I apply" and "what are your chances of getting in", I decided to embark on the task of finding the answer.
The following graphs represent the chances of getting into graduate programs based on how many programs you apply into. Of course, this is assuming that you are an average candidate for each of the schools, with the application-pool-average GRE scores, research experience, recommendation letters, and personal statements.
Live graph: https://www.desmos.com/calculator/inqypbtubg
Formula 1 is binomial distribution.
It represents the chance of getting into at least 1 school (y),
based on the number of schools applied (x).
This is assuming the average acceptance rate of x schools is 5%.
You can see that to have greater than 50% chance of getting into a school,
you would need to apply to at least 14 schools.
15% is the chance of getting into at least one of the top four best clinical psychology programs in California (given the average of acceptance rate of 3.88%): UCB, UCSD, UCLA, and USC.
Absolute chances of getting into at least one program when applying to the top 10 clinical psychology programs is about 30% (3.4% average per school). You can see that to get even a 50% chance, you would need to apply to about 20 such schools!
UCB, CA
UCLA, CA
SDSU/UCSD, CA
USC, CA
U Penn, PA
Penn State, PA
Yale, CT
Duke, NC
Stony Brook, NY
Rutgers, NJ