Question

upposea components of system all act independently, and each fails within a year 10% of the time Find the probability none fails within a year Find the probability at least one fails within a year Find the probability all fail within a year. Find the probability at least one works the entire year.

Answer 1

Here number of components are n=4, probability that a component will fail is p=0.1

X: Number of failures. Then X~ Binomial(n=4, p=0.1)

The Probability Mass Function of Binomial distribution is

1 ) The probability that none fail within a year is'

  

2) The probability that at least 1 fails within a year is

3) The probability that all fails within a year is

Pr(X = 4) = (: ) 0.14 x 0.94 | ) 0.14 . O.94-4-0.14 = 0.0001 Pr(X 4)

4)   Here we use p=1-0.1=0.9 and X: Number of working components.

The probability that at least 1 works entire year is