Answer to Question #112504 in Statistics and Probability for Bernie

Question

On a given day, each computer in the lab has at most one crash. There is a 5% chance
that a computer has a crash during the day, independent of the performance of any
other computers in the lab. Find the probability that on any given day, there are;
(a) exactly 3 crashes
(b) at most 3 crashes
(c) at least 3 crashes
(d) more than 3 and less than 6

Answer 1

Assume n - number of computers,k - number of crashed computers according to a bernoulli formula:

a)P_n(k=3)=C(n,3)*0.05³*0.95^(n-3)

b)P_n(k$\leq$3)=P_n(0)+P_n(1)+P_n(2)+P_n(3)=C(n,0)*0.05⁰*0.95ⁿ+C(n,1)*0.05¹*0.95^n-1+C(n,2)*0.05²*0.95^n-2+C(n,3)*0.05³*0.95^n-3

c)Pn(k$\geq$ 3)=P_n(3)+....+P_n(n)=C(n,3)*0.05³*0.95^n-3+...+C(n,n)*0.05ⁿ

d)Pn(3<k<6)=P_n(4)+P_n(5)=C(n,4)*0.05⁴*0.95^n-4+C(n,5)*0.05⁵*0.95^n-5