the probability that each will fail is .48
the probability that each will not fail is 1 - .48 = .52
the probability that at least 2 of the 5 will fail is equal to the probability that exactly 2 plus 3 plus 4 plus 5 will fail.
this is equal to 1 minus (the probability that exactly 0 plus 1) will fail.
either one of these should get you the correct answer.
since 1 minus ((the probability that exactly 0 plus 1) will fail requires less computations, then it's easier to go with that.
this is a binomial probability formula, which is:
p(x) = p^x * q^(n-x) * c(n,x)
p = .48
q = .52
n = 5
x = either 0,1,2,3,4, or 5.
p(0) = .48^0 * .52^5 * c(5,0)
p(1) = .48^1 * .52^4 * c(5,1)
c(n,x) = n! / (x! * (n-x)!)
consequently:
c(5,0) = 1
c(5,1) = 5
the formulas become:
p(0) = .48^0 * .52^5 * 1
p(1) = .48^1 * .52^4 * 5
this results in:
p(0) = .0380204032
p(1) = .175478784
p(0) + p(1) = .2134991872
1 - (p(0) + p(1)) = .7865008128
1 - (P(0) + p(1)) should be the same answer as p(2) + p(3) + p(4) + p(5)
the following excel spreadsheet shows you this is true.
my manually calculated numbers are:
p(0) + p(1) = .2134991872
1 - (p(0) + p(1)) = .7865008128
the excel spreadsheet number are:
p(0) + p(1) = 0.213499187
1 - (p(0) + p(1)) = 0.786500813
excel number are to 9 decimal places.
my numbers are to 10 decimal places.
round my numbers to 9 decimal places and you get:
p(0) + p(1) = .213499187
1 - (p(0) + p(1)) = .786500813
these exactly match the excel numbers when both are rounded to the 9 decimal places.