There are twelve machines in a factory The machines operate

There are twelve machines in a factory. The machines operate overnight while there are no workers in the factory. Records suggest the probability that each machine still works the next day is 0.7 and is independent of the other machines’ operating status. If all twelve machines are left working overnight, let P be the probability that four or more will still be working the following morning. In percent, in which interval does P lie?

Solution

Binomial Distribution

PMF of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
Where   
k = number of successes in trials
n = is the number of independent trials
p = probability of success on each trial

P( X < 4) = P(X=3) + P(X=2) + P(X=1) + P(X=0)
= ( 12 3 ) * 0.7^3 * ( 1- 0.7 ) ^9 + ( 12 2 ) * 0.7^2 * ( 1- 0.7 ) ^10 + ( 12 1 ) * 0.7^1 * ( 1- 0.7 ) ^11 + ( 12 0 ) * 0.7^0 * ( 1- 0.7 ) ^12
= 0.0017
P( X > = 4 ) = 1 - P( X < 4) = 0.9983