There are 5 servers. Each server has 1% downtime. What's the probability that at at least three servers are down? There are 5 servers. Each server has $1$% downtime. What's the probability that at at least three servers are down?
My reasoning is the following:
A) There is $(1-0.01)^5$ probability that $5$ from $5$ servers are up 
B) Thus, there is $1-(1-0.01)^5$ chance that at least one of $5$ servers is down
C) Similarly there is $1-(1-0.01)^4$ chance that at least one of any $4$ servers is down
D) And $1-(1-0.01)^3$ chance that at least one any $3$ servers is down
Now we can use the points above to calculate the answer:
E) Using B and C we can say that there is $(1-(1-0.01)^5)\cdot(1-(1-0.01)^4$ probability that at least $2$ of them are down (assuming at least one of servers is down, there is $(1-(1-0.01)^4$ probability that one of the rest 4 servers is down as well)
F) Finally, using D and E, there is $(1-(1-0.01)^5)\cdot(1-(1-0.01)^4)\cdot(1-(1-0.01)^3)$ probability that at least $3$ of them are down.
I saw a different answer $(1-\binom{5}{3}\cdot0.01^3)$ in one broadcast, which makes me think that my point E and F are wrong. Could somebody say if any of those answers is correct, and why?

Probability of at least three servers failing is pr(3) + pr(4) + pr(5) 
Probability of exactly A servers failing is 
Pr(A) is (5 Choose A) (1%) ^ A * (99%) ^ (5-A) 
so if you sum up pr(3) + pr(4) + pr(5) = 0.000009801 + 4.95E-08 + 1.05154E-10 = 0.0009851%
