Question

Three times each day a quality control engineer samples a component from a recently manufactured batch and tests it. Each part is given one of three classifications: conforming suitable for its intended use downgraded unsuitable for its intended purpose, but usable for another purpose S scrap - not usable An experiment consists of recording (in order) the classifications of the three parts tested on a particular day. List all of the outcomes in the sample space. (For example, two possible outcomes are CCS and DSC. CCS means that the first two parts were conforming, and the third was scrap. DSC means that the first was downgraded, the second was scrap, and the third was conforming. Note that this is not the same as CSD.) How many are there?
Answer all parts please

Answer 1

Define the events $C=\textup{conforming},D=\textup{downgraded},C=\textup{scap}$ . The sample space in the experiment $S$ has

$3^3=27$ elements.

a)The set  of elements that all the parts fall into the same category is $A=\left \{ CCC,SSS,DDD \right \}$
The number of elements that all the parts fall into the same category is the cardinality of the set ${\color{Blue} \left |A \right |=3}$

b)The set  of elements that at least 2 parts are conforming is $B=\left \{ CCC,CCS,CCD,CSC,CDC,DCC,SCC \right \}$
The elements that at least 2 parts are conforming is the cardinality of the set ${\color{Blue} \left |B \right |=7}$

c) The union of the sets $A,B$ is

$A\cap B=\left \{ CCC,SSS,DDD \right \}\cap \left \{ CCC,CCS,CCD,CSC,CDC,DCC,SCC \right \}\\ {\color{Blue} A\cap B=\left \{ CCC \right \}}$

d) The intersection $A\cap B'$ can be found as follows.

$A\cap B'+A\cap B=A\Rightarrow A\cap B'=A-A\cap B'$

Applying the above rule ( - represents the difference between the sets),

$A\cap B'=\left \{ CCC,SSS,DDD \right \}-\left \{ CCC \right \}\\ {\color{Blue} A\cap B'=\left \{ SSS,DDD \right \}}$