Question

Three components are connected to form a system as shown in the accompanying diagram. Because the components in the 2–3 subsystem are connected in parallel, that subsystem will function if at least one of the two individual components functions. For the entire system to function, component 1 must function and so must the 2–3 subsystem.

The experiment consists of determining the condition of each component (S(success) for a functioning componentand F (failure) for a non-functioning component).

a. Which outcomes are contained in the event Athat exactly two out of the three components function?

b. Which outcomes are contained in the event Bthat at least two of the components function?

c. Which outcomes are contained in the event Cthat the system functions?

d. List outcomes in C’, A \( \cup \)C, A \( \cap \)C, B \( \cup \)C, and B \( \cap \)C.

Short answer

a. The possible outcomes are:

\(A = \left\{ {SSF,SFS,FSS} \right\}\).

b. The possible outcomes are:

\(B = \left\{ {SSS,SSF,SFS,FSS} \right\}\).

c. The possible outcomes are:

\(C = \left\{ {SSF,SFS,SSS} \right\}\).

d. The outcomes are:

\(C' = \left\{ {FFF,FFS,FSS,FSF,SFF} \right\}\)

\(A \cup C = \left\{ {SSF,SFS,FSS,SSS} \right\}\)

\(B \cup C = B\)

\(A \cap C = \left\{ {SSF,SFS} \right\}\)

\(B \cap C = C\)

Step-by-step solution

Given information

The number of components that are connected to form a system are 3.

Components 2 and 3 are connected in parallel. The subsystem will work if at least one of the two individual components functions.

S represents success and F represents Failure.

List of the possible outcomes

a.

Event A represents that exactly two out of the three components function.

The possible outcomes of A is given as,

\(A = \left\{ {SSF,SFS,FSS} \right\}\)

List of the possible outcomes

b.

Event B represents that at least two of the components function.

The possible outcomes of B is given as,

\(B = \left\{ {SSS,SSF,SFS,FSS} \right\}\)

List of the possible outcomes

c.

Event C represents the system functions.

The possible outcomes of C is given as,

\(C = \left\{ {SSF,SFS,SSS} \right\}\)

List of the possible outcomes in different events

d.

Referring to parts a, b, and c,

\(A = \left\{ {SSF,SFS,FSS} \right\}\)

\(B = \left\{ {SSS,SSF,SFS,FSS} \right\}\)

\(C = \left\{ {SSF,SFS,SSS} \right\}\)

The complementary of an event C consists of all the outcomes that are not contained in C.

The possible outcomes are,

\(C' = \left\{ {FFF,FFS,FSS,FSF,SFF} \right\}\)

A union of two events A and C consists of all the outcomes that are either in A or C or in both events

\(A \cup C = \left\{ {SSF,SFS,FSS,SSS} \right\}\)

Similarly, for events B and C, we have,

\(\begin{aligned}B \cup C &= \left\{ {SSS,SSF,SFS,FSS} \right\}\\ &= B\end{aligned}\)

The intersection of two events A and C consists of all the outcomes that are present in both events.

\(A \cap C = \left\{ {SSF,SFS} \right\}\)

Similarly for events B and C, we have,

\(\begin{aligned}B \cap C &= \left\{ {SSF,SFS,SSS} \right\}\\ &= C\end{aligned}\)