Question

As engineers, we endeavor to improve or increase the reliability of the systems or components that we design. In that sense, reliability is the probability of a system or component meeting its requirements successfully. A component with 80% reliability has a probability of p=.8 of meeting it's requirements. As a result, we can see reliability as a specialized application of probability theory.

Use the rules of probability to explain how you would calculate the overall reliability of one system that has three components that have to perform in series (and so all components must perform to have a successful outcome) versus another system that has three components that operate in parallel (and so only one of the three needs to perform to have a successful outcome).

of probability theory.

Use the rules of probability to explain how you would calculate the overall reliability of one system that has three components that have to perform in series (and so all components must perform to have a successful outcome) versus another system that has three components that operate in parallel (and so only one of the three needs to perform to have a successful outcome).

Answer 1

Consider the reliablity of one system that has three components A,B amd C with reliability R_A. R_B R_C respectively.

(1) Consider componenets A, B and C be in series.i.e., All the three components must perform to have a successful outcome.

So, by Multiplicatio Theorem of probability:

$R_{_{S}}=R_{_{_{_{A}}}}\times R_{_{B}}\times R_{_{C}}$ ,

where R_S is the reliability of the overall system.

This is the formula for overall reliability R_S of one system that has three components A,B and C that have to perform in series.

(2) Consider components A, B and C be in parallel. i.e., only one of the three A,B or C needs to perform to have a successful outcome.

So,

the overall system fails when all the three components A , B and C fail.

Now,

probability of failure of A = 1 - R_A

Probability of failure of B = 1 - R_B

Probability of failure of C = 1 - R_C

By multiplication Theorem,

Probability of failure of overall system =

$1-R_{_{_{S}}}=(1-R_{_{A}})\times (1-R_{_{_{B}}})\times (1-R_{C})$

Thus, reliability R_S of overall system is given by:

$R_{_{_{S}}}=1-(1-R_{_{A}})\times (1-R_{_{_{B}}})\times (1-R_{C})$

This is the formula for overall reliability of one system that has three components that operate in parallel.