4.1-4.2 Sample Spaces and Probability

 Should I carry an umbrella today?
 Will my car battery last until spring?
 Should I accept that new job?

 The chance of an event occurring.
 Examples: card games, slot machines,
lotteries, …insurance, investments, weather
forecasting
 Basis of inferential statistics

 Probability Experiment: A chance process
that leads to well-defined results called
outcomes.
 Outcome: The result of a single trial of a
probability experiment.
 Trial: one flip of a coin, one roll of a die, etc.
 Sample Space: The set of all possible
outcomes of a probability experiment.

Die 1
Die 2
1 2 3 4 5 6
1 (1,1)
2 (1,2)
3 (1,3)
4
5
6

Die 1
Die 2
1 2 3 4 5 6
1 (1,1) (2,1) (3,1) (4,1) (5,1) (6,1)
2 (1,2) (2,2) (3,2) (4,2) (5,2) (6,2)
3 (1,3) (2,3) (3,3) (4,3) (5,3) (6,3)
4 (1,4) (2,4) (3,4) (4,4) (5,4) (6,4)
5 (1,5) (2,5) (3,5) (4,5) (5,5) (6,5)
6 (1,6) (2,6) (3,6) (4,6) (5,6) (6,6)

H A 2 3 4 5 6 7 8 9 10 J Q K
D A 2 3 4 5 6 7 8 9 10 J Q K
S A 2 3 4 5 6 7 8 9 10 J Q K
C A 2 3 4 5 6 7 8 9 10 J Q K
52 Possible Outcomes

Kid 3Kid 2Kid 1
Boy
Boy
Boy
Girl
Girl
Boy
Girl
Kid 3Kid 2Kid 1
Girl
Boy
Boy
Girl
Girl
Boy
Girl
8
Possibilitie
s with
Three
Children

 A Tree Diagram is a device consisting of line
segments emanating from a starting point
and also from the outcome point. It is used
to determine all possible outcomes of a
 An Event consists of a set of outcomes of a

 Simple Event: an event with one outcome
(rolling a die one time, choosing one card)
 Compound Event: an event with more than
one outcome (rolling an odd number on one
die -3 possibilities)

 Classical
 Empirical (Relative Frequency)
 Subjective

 Uses sample spaces to determine numerical
probability that an event will happen.
 An experiment is not performed to determine
the probability of an event.
 Assumes that all outcomes in a sample
space are equally likely to occur (6
possibilities on a die have equally likely
chance of occurring)

 Probability of any event E is
Number of outcomes in E .
Total number of outcomes in the sample space
 This probability is denoted by
P(E) = n(E)
n(S)
 Answers given as fractions, decimals or
percentages.

 Reduced fractions or decimals rounded to
two or three decimal places
 If probability is extremely small, round the
decimal to the first nonzero digit after the
decimal point. (0.000000478 = 0.0000005).

 And means “at the same time.”
 Or means
› Inclusive or (drawing a queen or a heart means
looking for one of 4 queens or one of 13 hearts.
Q of H included in both sets, so possibilities are
4 + 13 -1 = 16)
› Exclusive or (drawing a queen or a king means
looking for one of 4 queens or one of 4 kings. 4
+ 4 = 8 possibilities).

 A card is drawn from a standard deck. Find
these probabilities:
› A) Of getting a 10.
› B) Of getting the 5 of clubs (a 5 and a club)
› C) Of getting a 7 or a heart
› D) Of getting an Ace or a 2

1. The probability of any event E is a number
(either a fraction or a decimal) between and
including 0 and 1. This is denoted by 0 ≤
P(E) ≤ 1.
2. If an event E cannot occur (the event
contains no members in the sample space),
its probability is 0.

3. If an event E is certain, then the probability
of E is 1.
4. The sum of the probabilities of all the
outcomes in the sample space is 1.

 The Complement of event E is the set of
outcomes in the sample space that are not
included in the outcomes of event E. The
complement of E is denoted by Ē (E “Bar”).
 Find the complement of selecting a letter of
the alphabet and getting a vowel.

P(Ē) = 1 – P(E) or P(E) = 1 - P(Ē) or
P(E) + P(Ē) = 1

 Used to pictorally represent the probability of
events.
 Venn Diagram for the probability and
complement:
P(S) = 1
P(E)
P(Ē)
P(E)

 The type of probability that uses frequency
distributions based on observations to
determine numerical probabilities of events.
 For example, one might actually roll a given
die 6,000 times to observe the frequencies of
each possibility. They would then use the
outcomes of the experiment upon which to
base their probability.

 Given a frequency distribution, the
probability of an event being in a given class
is
P(E) = Frequency for class = f .
Total frequencies in the distribution n
 This probability is called empirical
probability and is based on observation.

 For a recent year, 51% of the families in the US
had no children under the age of 18; 20% had
one child; 19% had two children; 7% had three
children; and 3% had four or more children. If a
family is selected at random, find the probability
that the family has
› Two or three children
› More than one child
› Less than three children
› Based on the answers in the first three parts, which is
most likely to occur?

 When a probability experiment is repeated a
large number of times, the relative frequency
probability of an outcome will approach its
theoretical probability.

 The type of probability that uses a probability
value based on an educated guess or
estimate, employing opinions and inexact
information.

4.1-4.2 Sample Spaces and Probability

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