a machine has 7 independent components the probability that

a machine has 7 independent components, the probability that a component fails is .2. in order for the machine to operate at least 6 bits of its 7 components must work. find the probability that the machine fails?

Solution

Here,

P(fail) = 1 - P(works) = 1 - P(at least 6 work)

Or, that is,

P(fail) = P(at least 2 fail)

Note that P(at least x) = 1 - P(at most x - 1).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    7      
p = the probability of a success =    0.2      
x = our critical value of successes =    2      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   1   ) =    0.5767168
          
Thus, the probability of at least   2   successes is  
          
P(at least   2   ) =    0.4232832 [ANSWER]