The effect depends on many factors. It’s not just a simple problem where if you want to go to a 92% rejection college, then a high stat applicant has a 1 - 0.92^20 = 81% chance of getting admitted somewhere, if you apply to 20 schools. Instead the decisions are often well correlated with oneanother. If something non-stat about your application is weak compared to other applicants, you may be highly likely to be rejected everywhere.
Different colleges look for and emphasize different criteria, so the specific chance of admission will vary by college, but it may be more desirable to choose colleges that are an especially good fit and./or you believe your chance of admission may be relatively larger, rather than applying to a huge number of colleges with little filter besides choosing colleges that are highly ranked/selective.
Another relevant issue is differences in quality of application as number of applications increases, which may change chance of acceptance for that school. For example, if a student who applies to 20 schools might put less average effort in to each application that a student who applies to 4 schools. The could include things like being less likely to show demonstrated interest at schools that consider this, or things like cut and pasting essays, perhaps sometimes forgetting to change the college name when pasting. Alternatively a student could also better master the process on later applications, such as getting used to interviews and doing better at them as he/she gains more experience. Again it depends on a variety of factors.
One can look at real anecdotal outcomes rather than theory by looking at decision outcomes threads in this forum, such as the one at Can more graduated seniors do "actual results" threads? - #2340 by Betseyt . For example, the most recent poster who applied to near 20 colleges had the following results. The acceptances occurred at colleges that tend to emphasize stats/rank, and the waitlists/rejections generally occurred at colleges that tend to place less relative emphasis on stats. I expect most of the acceptances could have been predicted well. Similarly I expect the poster could have got other acceptances at other colleges, had he applied to others that tend to emphasize stats, and place less emphasis on the application criteria he rated as his weaker arees.
Texas Resident planning to study CS
Stats – 4.0 with 14 APs
SAT – 1480
ECs – CS project with 10k+ users, Published research, Robotics, Varsity Track
LORs – Poster estimated as 5/10, 7/10, and 8/10
Essays – Poster estimated as 5/10Attending – UC Berkeley
Accepted – UC Berkeley, Texas A&M, UIUC (EA),USC, UT Austin
Waitlisted – Cornell, CMU, Purdue, GeorgiaTech (EA), Michigan, Rice
Rejected – Columbia, Dartmouth, Duke, Harvard, Northwestern, Penn, Stanford (REA)