This post covers Introduction to probability from Statistics for Engineers and Scientists by William Navidi.

Exercises

• Items are inspected for flaws by two quality inspectors. If a flaw is present, it will be detected by the first inspector with probability 0.9, and by the second inspector

  with probability 0.7. Assume that both inspectors inspect every item and that if an item has no flaw, then neither inspector will detect a flaw.

  a. If an item has a flaw, what is the probability that it will not be found by first inspector?

  b. If an item has a flaw, what is the probability that it will not be found by both inspectors?

Solution: (a) 0.011 (b) 0.0033

• A system contains two components, $C$ and $D$, connected in parallel as shown in the diagram.

  

  Assume $C$ and $D$ function independently. For the system to function, either $C$ or $D$ must function.

  a. If the probability that $C$ fails is $0.08$ and the probability that $D$ fails is $0.12$, find the probability that the system functions.

  b. If both $C$ and $D$ have probability $p$ of failing, what must the value of $p$ be so that the probability that the system functions is $0.99$?

  c. If three components are connected in parallel, function independently, and each has probability $p$ of failing, what must the value of $p$ be so that the probability that the system functions is $0.99$?

  d. If components function independently, and each component has probability $0.5$ of failing, what is the minimum number of components that must be connected in parallel so that the probability that the system functions is at least $0.99$?

  Solution: (a) 0.9904 (b) 0.1 (c) 0.2154 (d) $n=7$.