### 1.2 Sample spaces & Events

Def: Sample space: set of all possible outcomes of an experiment; elements called sample points
• usually denote by S
• must include all possible outcomes
• sometimes more than one possibility for S, depending on how outcomes are specified
ex:
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.)

ex:
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!!)

Def: An event is any subset of sample space (i.e., any set of possible outcomes) - can consist of a single element

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}

Notes:
• 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
ex: (rocket)
if rocket blows up during 2nd stage; then event A above has occurred, event B hasn't

Can consider union, intersection, complement of events to get new events
• AB:  either A occurs, or B occurs, or both
• AB:  both A & B occur
• A':  A does not occur
ex: (rocket)
AB = event rocket fails and gets thru second stage = {ssf, sssf}
A' = event rocket doesn't fail = {ssss}
Def: Say events A & B are mutually exclusive if  AB =
• 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.

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