Answer to Question #323802 in Statistics and Probability for Bless

Question

A complex electronic system is built with a certain number of backup components in its subsystems. One subsystem has four identical components, each with a probability of .2 of failing in less than 1000 hours. The subsystem will operate if any two of the four components are operating. Assume that the components operate independently. Find the probability that;

(a) Exactly two of the four components last longer than 1000 hours.

(b) The subsystem operates longer than 1000 hours.

Answer 1

$X-number\,\,of\,\,working\,\,components:\\X\sim Bin\left( 4,0.8 \right) \\a:\\P\left( X=2 \right) =C_{4}^{2}\cdot 0.8^2\cdot 0.2^2=0.1536\\b: 2\ or\ more\ components\ work\ longer\ than\ 1000\ hours\\P\left( X\geqslant 2 \right) =1-P\left( X=0 \right) -P\left( X=1 \right) =\\=1-C_{4}^{0}\cdot 0.8^0\cdot 0.2^4-C_{4}^{1}\cdot 0.8^1\cdot 0.2^3=0.9728$