Problem $01$

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 $54M\frac{b}{s},$ medium $(m)$ at $11M\frac{b}{s},$ and low $(l)$ at $1M\frac{b}{s}.$ 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 A$1$ be the event "medium speed connection." What are the outcomes in A$1?$

c$)$ Let A$2$ be the event "mouse click." What are the outcomes in A$2?$

d$)$ Let A$3$ be the event "high speed connection or low speed connection." What are the outcomes in $A3?$

e$)$ Are $A1,A2,$ and $A3$ mutually exclusive?

f$)$ Are $A1,A2,$ and $A3$ collectively exhaustive?

Problem $02$

Shuffle a deck of cards and turn over the first card.

a$)$ What is the sample space of this experiment?

b$)$ How many outcomes are in the event that the first card is a heart?