Answer
a)
The outcomes in the sample space are:
\[S=\left\{ \begin{align}
& (0,0,0,0),\text{ }(0,0,0,1),\text{ }(0,0,1,0),\text{ }(0,1,0,0),\text{ } \\
& (1,0,0,0),\text{ }(1,0,0,1),\text{ }(0,1,0,1),\text{ }(0,0,1,1), \\
& (1,1,1,0),\text{ }(0,1,1,0),\text{ }(1,1,0,1),\text{ }(1,0,1,1) \\
& (1,1,1,0),\text{ }(0,1,1,1),\text{ }(1,0,1,0),\text{ }(1,1,1,1) \\
\end{align} \right\}\]
b)The outcomes making up the event that exactly two phones are being used is:
\[A=\left\{ (1,0,0,1),\text{ }(0,1,0,1),\text{ }(0,0,1,1),\text{ (1,1,0,0), }(0,1,1,0),\text{ }(1,0,1,0) \right\}\]
c) There are 1+k outcomes allowing for the possibility that at most one more call could be received.
Work Step by Step
a)
The telemarketer’s “bank” is comprised of 4 telephones. Let 0 indicates that the phone is available and 1 indicates that a caller is on the line.
Hence there are 16 possible combinations in the sample space.
Therefore, the outcomes of the sample space S are:
\[S=\left\{ \begin{align}
& (0,0,0,0),\text{ }(0,0,0,1),\text{ }(0,0,1,0),\text{ }(0,1,0,0),\text{ } \\
& (1,0,0,0),\text{ }(1,0,0,1),\text{ }(0,1,0,1),\text{ }(0,0,1,1), \\
& (1,1,0,0),\text{ }(0,1,1,0),\text{ }(1,1,0,1),\text{ }(1,0,1,1) \\
& (1,1,1,0),\text{ }(0,1,1,1),\text{ }(1,0,1,0),\text{ }(1,1,1,1) \\
\end{align} \right\}\]
(b)
Let the event A represent that exactly two phones are being used. So we have to choose two 0’s from sample space S. We see that six of the sample outcomes in S constitute the event A.
Therefore, the outcomes of the event A are:
\[A=\left\{ (1,0,0,1),\text{ }(0,1,0,1),\text{ }(0,0,1,1),\text{ (1,1,0,0), }(0,1,1,0),\text{ }(1,0,1,0) \right\}\]
(c)
At most one more call could be received; this means that we have to choose three or four 1’s from sample space S.
There are 5 sample outcomes from the sample space allowing for the possibility that at most one more call could be received.
Therefore, the possible outcomes are
\[\left\{ (1,1,0,1),\text{ }(1,0,1,1),\text{ }(1,1,1,0),\text{ }(0,1,1,1),\text{ }(1,1,1,1) \right\}\]
In general, the telemarketer had k phones then there are 1+k outcomes allowing for the possibility that at most one more call could be received.
