4.1-4.2 sample spaces and probability
TRANSCRIPT
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
BoyBoy
Girl
GirlBoy
Girl
Kid 3Kid 2Kid 1
Girl
BoyBoy
Girl
GirlBoy
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
probability experiment.
An Event consists of a set of outcomes of a
probability experiment.
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.
p.185-187 #1-20
p.185-187 #21-36