A hypothetical wi-fi transmission can take place at any of three speeds depending on the condition of the radio channel between a laptop and an access point. The speeds are high (h) at 54 Mb/s, medium (m) at 11 Mb/s, and low (l) at 1 Mb/s. A user of the wi-fi connection can transmit a short signal corresponding to a mouse click (c), or a long signal corresponding to a tweet (t). Consider the experiment of monitoring wi-fi signals and observing the transmission speed and the length. An observation is a two-letter word, for example, a high-speed, mouse-click transmission is hm.
A) What is the sample space of the experiment?
B) Let A1 be the event "medium speed connection." What are the outcomes in A1?
C) Let A2 be the event "mouse click." What are the outcomes in A2?
D) Let A3 be the event "high speed connection or low speed connection." What are the outcomes in A3?
E) Are A1, A2 and A3 mutually exclusive? Are A1, A2, and A3 collectively exhaustive?

A) The sample space(S) = {hc, mc, lc, ht, mt, lt}

B) Outcome in A1 ={c, t}

C) Outcomes in A2 = {hc, mc, lc}

D) Outcomes in A3 = {hc, ht, lc, lt}

Ei) No, A1, A2 and A3 are not mutually exclusive.

Eii) Yes, A1, A2 and A3 are collectively exhaustive

Explanation:

A) First of all, from the question, there will be 6 elements in the sample space. These are;

- 3{high(h), medium (m), low(l)} which correspond to mouse click(c)

- 3{high(h), medium(m), low(l)} which corresponds to tweet (t).

Thus, we can say that the sample space(S) = {hc, mc, lc, ht, mt, lt}

B) For A1 to correspond to medium speed connection, it means it must have 2 outcomes which are; mouse click (c) and tweet (t)

Thus, A1 ={c, t}

C) For A2 to be the event mouse click (c), it will have 3 outcomes namely;

High click, medium click and low click.

So A2 = {hc, mc, lc}

D) For A3 to be the event "high speed connection or low speed connection, we will have 4 outcomes which are 2 each of high and low speed connection. These are ;

So A3 = {hc, ht, lc, lt}

E) i) For A1, A2 and A3 to be mutually exclusive, it will result in an empty set. This can be expressed as;

A1 ∩ A2 ∩ A3 = Φ

Now, looking at the results for A1, A2 and A3 above, we can see that no single item appears in the 3 of them at the same time.

Thus, A1 ∩ A2 ∩ A3 = Φ

E) ii) For A1, A2 and A3 to be collectively exhaustive, the result of A1 U A2 ∪ A3 has to contain every possible member in the sample space (S).

Thus, we can write it as A1 U A2 ∪ A3 = S

Now, A1 U A2 ∪ A3 = {hc, ht, mc, mt, lc, lt}

Since A1 U A2 ∪ A3 results in a set that contains all the members of the sample space, A1, A2 and A3 are collectively exhaustive

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