Question

Calls are repeatedly placed to a busy phone line until a connect is achieved. Let c = connect and b = busy, select the correct sample space, S, below: A. S = {b, c} B. S = {bbbbbb, ...} C. S = {cb, cbb, cbbb, ...} D. S = {c, bc, bbc, bbbc, ...}

Answer 1

Answer:

(D)

Step-by-step explanation:

$(A)$

Let's have a look at the first sample space, the set

                                      $S = \{b,c\}$

As we can see, it consists of two elements. Element $b$ implies that a call was made, but the line was busy, and the element $c$ implies that the call was connected. However, this sample space doesn't correspond to the event in which the calls are repeatedly made until a connect is achieved. The element $c$ is suitable, because it is possible that the line wasn't busy at the moment of calling, but the element  $b$ implies that after receiving the busy signal, there were no further attempts to connect to the line.

$(B)$

The second sample space is

                                  $S =\{bbbbbb, \ldots \}$

It corresponds to the case in which after a few attempts to connect to the line, the person gave up. Therefore, it doesn't correspond to our problem.

$(C)$

The third sample space is

                                  $S = \{ cb, cbb, cbbb, \ldots \}$

It corresponds to the case in which after one successful attempt, the person tried again for a few more times, and also gave up before connecting again. Therefore, it doesn't correspond to our problem either.

$(D)$

The last sample space is

                              $S = \{ c, bc, bbc, bbbc, \ldots\}$

The first element corresponds to the case that the line wasn't busy in the first attempt. The second element, $bc$, corresponds to the case in which the line was busy in the first attempt but it wasn't in the second. The third is the one in which the line was busy for three attempts, but wasn't in the fourth, and so on. As we can see, the common thing about all this elements is that the there were no further attempts after a connection was achieved, which corresponds to our case. Therefore, this is the correct sample space.