A large electronic office product contains 2000 electronic c

A large electronic office product contains 2000 electronic components. Assume that the probability that each component operates without failure during the useful life of the product is 0.995, and assume that the components fail independently. Approximate the probability that five or more of the original 2000 components fail during the useful life of the product.

Solution

The probability that it will fail (per component) is

p = 1 - 0.995 = 0.005.

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 =    2000      
p = the probability of a success =    0.005      
x = our critical value of successes =    5      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   4   ) =    0.02896924
          
Thus, the probability of at least   5   successes is  
          
P(at least   5   ) =    0.97103076 [ANSWER]