7-1 Sample Spaces, Events, and Probability
Random experiment:
a process whose outcome cannot be predicted
Sample space (of a random experiment):
the set of all possible outcomes of the experiment
Example:
Random experiment:
Sample space:
toss a coin twice
{HH, HT, TH, TT} (four possible
outcomes)
EVENTS
event has a specific technical meaning in probability theory
Example:
Random experiment: flip coin two times
Sample space: the set {HH, HT, TH, TT}
Two (of many possible) events:
(1) getting at least one head: E1 = {HH, HT, TH}
(2) getting no tails:
E2 = {HH}
Definition: an event is a subset of the sample space
 for the sample space of flipping a coin twice
 there are lots of events (subsets) (24 = 16, to be exact)
Note: for a sample space of size n, there are 2n possible events
An event is said to occur if any of its outcomes occurs.
7-1
p. 1
Events can sometimes be described in words
e.g., for the flipping-a-coin-twice experiment:
In words
no heads
one tail
at least one head
more than one head
Set notation
{TT}
{TH, HT}
{TH, HT, HH}
{HH}
PROBABILITIES OF OUTCOMES
Random experiment: flip coin twice
Sample Space: S = {HH, HT, TH, TT}
Question: What is the probability of getting HH?
Answer: there is 1 chance out of 4 of getting HH = 1/4
in the vocabulary and notation of probability theory:
 vocabulary:
 notation:
the probability of HH is 1/4
P(HH) = 1/4
for S = {HH, HT, TH, TT} we have:
P(HH) = ¼ P(HT) = ¼ P(TH) = ¼ P(TH) = ¼
Sample spaces of equally likely outcomes
the probability of any single outcome
in a sample space of n equally likely outcomes is
1/n
Notes:
 probability: a number between 0 and 1 inclusive
 sum of probabilities of all outcomes in a sample space: 1
7-1
p. 2
Using these ideas, we can move on to sample spaces where the
outcomes are not all equally likely.
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General probability spaces
start with a sample space S
assign a probability to each outcome
if each probability is between 0 and 1 inclusive, and …
they add up to 1
you have created a valid probability space
SAMPLE SPACES WITH UNEQUALLY LIKELY OUTCOMES
 consider shooting a basket
 SS = {hit, miss}
 the only way to assign probabilities to the outcomes is to
estimate them by experiment
 suppose I shoot 100 baskets, and make 35
 P(hit) = 35/100 = .35 P(miss) = 65/100 = .65
 is this a valid probability space?
PROBABILITIES OF EVENTS
probability of an event - the sum of the probabilities of the
individual outcomes in the event:
P({HH}) = ¼
P({HH, TT}) = P(HH)+P(TT) = ½
P() = 0
P({HH, HT, TH, TT}) = 1
7-1
p. 3
Experiment: Draw a card from a 52-card deck
Sample space = {1H, 1C, 1D, 1S, 2H, 2C, 2D, 2S, . . . }
Assumption: each card has same likelihood of being drawn - a
probability space of equally likely outcomes
Question 1: Probability of drawing King of Hearts = ?
Answer: Pr(KH) = 1 chance out of 52 = 1/52
Question 2: Probability of drawing an ace = ?
Event: drawing an ace = A = { AH, AS, AD, AC}
Pr(A) = 1/52 + 1/52 + 1/52 + 1/52 = 4/52 = 1/13
 the probability of an event is the sum of the probabilities of its
individual outcomes
 or think of it as 4 chances out of 52
Probabilities of events in
spaces of equally likely outcomes
In a space
of n equally likely outcomes
the probability of an event
with m outcomes is
m/n
7-1
p. 4