Three telephone lines are monitored. At a given time, each telephone line can be

Question

Three telephone lines are monitored. At a given time, each telephone line can be in one of the following three modes: (1) voice mode, i.e., the line is busy and someone is speaking (2) data mode, i.e., the line is busy with a modem of fax signal, and (3) inactive mode, i.e., the line is not busy. We denote the three modes as V , D, and I respectively. For example, if the rst and second lines are in data mode and the third line is in inactive mode, the observation is DDI.

What would the

a) Write the elements of the event A = {at least two voice modes}. How would you calculate the total number of combinations? I know its 7 total possible from the solution but I just dont know how you figure this out.

b) Write the elements of the event B = {number of data modes > 1 + number of voice modes}.

Explanation / Answer

A) A has at least two voice modes, which can be written as A has two voice modes or A has three voice modes.

Number of ways in which two voice modes can occur in 3 = 2 x 3C2

(here 2 represents the number of options available for the third mode and 3C2 represents the number of arrangements of 2Vs possible)

= 2 x 3 = 6

Number of ways in which 3 out of 3 voice modes can occur = 1

So, A = {VVD, VVI, VDV, VIV, DVV, IVV, VVV}

B) if number of voice modes = 0, number of data modes can be 2 or 3

If number of voice modes = 1, number of data modes = 3

So, the possible cases are, (0voice, 2 data and 1 inactive) + (0 voice and 3 data)

= {DDI, DID, IDD, DDD}

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