问题及解答

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 component and $F$ (failure) for a nonfunctioning component].

A. What outcomes are contained in the event $A$ that exactly two out of the three components function?
B. What outcomes are contained in the event $B$ that at least two of the components function?
C. What outcomes are contained in the event $C$ that the system functions?
D. List outcomes in $C^{c}$, $A\cup C$, $A\cap C$, $B\cup C$, and $B\cap C$.
 

Reference:

Jay L. Devore, Probability and Statistics, For Engineering and The Sciences (Fifth Edtion)

A.

The event $A$ is the subset which consists of the following elements:

1 2 3
S S F
S F S
F S S

That is, $A=\{SSF, SFS, FSS\}$

B.

Since event $B$ is consisted by the element of that at least two components work, every element in $A$ is also belong to $B$. Besides, SSS is contained in $B$. Thus 

\[
B=A\cup\{SSS\}=\{SSS, SSF, SFS, FSS\}
\]

C

$C$ is the event of elements that make system work. According the question, the system function if and only if the component 1 works and the subsystem works. While the subsystem works if at least one of the two components (2 and 3) work. Thus, 

1 2 3
S S S
S S F
S F S

That is, 

\[
C=\{SSS, SSF, SFS\}
\]

Note that $C\subset B$.

D.

The complementary set of $C$ is

\[
C^{c}=\{FFF, FFS, FSF, SFF, FSS\}
\]

\[
A\cup C=\{SSF, SFS, FSS, SSS\}
\]

\[
A\cap C=\{SFS, SSF\}
\]

Since $C$ is a subset of $B$, we have

\[
B\cup C=B,\quad B\cap C=C.
\]