1.2 Sample spaces & Events

• usually denote by S
• must include all possible outcomes
• sometimes more than one possibility for S, depending on how outcomes are specified

flip coin 3 times; sample space S is
S = {hhh, hht, hth, htt, thh, tht, tth, ttt}   (8 possible outcomes)
or
S = {3 heads, 2 heads & 1 tail, 1 head & 2 tails, 3 tails}   (4 possible outcomes)
 
either is acceptable as the sample space; which one is used might depend on what we're interested in investigating. (The first has a very nice property not shared by the second: each of the outcomes is equally likely to occur! Because of this, we'll usually use the first as our sample space.)
 

Have 4 stages of a rocket; any one can fail, at which point mission is over. A logical sample space representing all possible outcomes would be
S = {f, sf, ssf, sssf, ssss},
where ssf represents the outcome in which the first two stages succeed but the third fails. (Hopefully, outcomes not equally likely!!)
 
 

ex: (rocket)

The event that the rocket fails at some stage is subset A = {f, sf, ssf, sssf}
The event that rocket goes through 2nd stage is subset B = {ssf, sssf, ssss}
 

• The empty set   is a subset, hence an event; called the impossible event
• The entire sample space  S  is a subset, hence an event; called the certain event
• When the actual outcome of the experiment is a member of the subset, we say the event has occurred

if rocket blows up during 2nd stage; then event A above has occurred, event B hasn't
 

• AB:  either A occurs, or B occurs, or both
• AB:  both A & B occur
• A':  A does not occur

AB = event rocket fails and gets thru second stage = {ssf, sssf}
A' = event rocket doesn't fail = {ssss}

• A and B have no outcomes in common
• events A and B can't both occur simultaneously

ex: (rocket)

Let C = event rocket succeeds; then A & C are mutually exclusive. A & B aren't; they share the outcomes ssf and sssf. In other words, events A and B could both occur, if in the actual outcome the rocket fails, but gets through the second stage.