Question 1203227
p(x) = p^x * q^(n-x)*c(n,x)


n = 14
x = 0 to 14
p = .53
q = 1 minus p = .47



probability of getting exactly 12 successes is p(12) = .53^12 * .47^(14-12) * c(14,12) = 0.009875237.


probability of getting more than 12 successes is equal to p(13) + p(14) = 0.001713216 + 0.000137995 = .001851211.


probability of getting less than or equal to 12 successes is 1 minus the probability of getting more than 12 successes is equal to 1 minus p(13) minus p(14) = 1 minus 0.001713216 minus 0.000137995 = .998148789.


here's what it looks like in excel.


<img src = "http://theo.x10hosting.com/2023/080803.jpg">


total probbility is equal to 1, as it should.