Transcript Document

Lesson 6 – 2a
Probability Models
Knowledge Objectives
• Explain what is meant by random phenomenon.
• Explain what it means to say that the idea of
probability is empirical.
• Define probability in terms of relative frequency.
• Define sample space.
• Define event.
Knowledge Objectives Cont
• Explain what is meant by a probability model.
• List the four rules that must be true for any
assignment of probabilities.
• Explain what is meant by equally likely outcomes.
• Define what it means for two events to be
independent.
• Give the multiplication rule for independent events.
Construction Objectives
• Explain how the behavior of a chance event differs in
the short- and long-run.
• Construct a tree diagram.
• Use the multiplication principle to determine the
number of outcomes in a sample space.
• Explain what is meant by sampling with replacement
and sampling without replacement.
• Explain what is meant by {A  B} and {A  B}.
• Explain what is meant by each of the regions in a
Venn diagram.
Construction Objectives Cont
• Give an example of two events A and B where A  B
= .
• Use a Venn diagram to illustrate the intersection of
two events A and B.
• Compute the probability of an event given the
probabilities of the outcomes that make up the
event.
• Compute the probability of an event in the special
case of equally likely outcomes.
• Given two events, determine if they are independent.
Vocabulary
• Empirical – based on observations rather than
theorizing
• Random – individuals outcomes are uncertain
• Probability – long-term relative frequency
• Tree Diagram – allows proper enumeration of all
outcomes in a sample space
• Sampling with replacement – samples from a
solution set and puts the selected item back in
before the next draw
• Sampling without replacement – samples from a
solution set and does not put the selected item back
Vocabulary Cont
• Union – the set of all outcomes in both subsets
combined (symbol: )
• Empty event – an event with no outcomes in it
(symbol: )
• Intersect – the set of all in only both subsets
(symbol: )
• Venn diagram – a rectangle with solution sets
displayed within
• Independent – knowing that one thing event has
occurred does not change the probability that the
other occurs
• Disjoint – events that are mutually exclusive (both
cannot occur at the same time)
Idea of Probability
Chance behavior is unpredictable in the short run, but
has a regular and predictable pattern in the long run
The unpredictability of the short run entices people to
gamble and the regular and predictable pattern in the
long run makes casinos very profitable.
Randomness and Probability
We call a phenomenon random if individual outcomes
are uncertain but there is nonetheless a regular
distribution of outcomes in a large number of
repetitions
The probability of any outcome of a random
phenomenon is the proportion of times the outcome
would occur in a very long series of repetitions.
That is, probability is long-term frequency.
Example 1
Using the PROBSIM application on your calculator flip
a coin 1 time and record the results? Now flip it 50
times and record the results. Now flip it 200 times
and record the results. (Use the right and left arrow
keys to get frequency counts from the graph)
Number of Rolls
1
Heads
0
Tails
1
51
18
33
251
117
134
Probability Models
Probability model is a mathematical description of a
random phenomenon consisting of two parts: a
sample space S and a way of assigning probabilities to
events
S
E
1
F
5
3
2
6
4
Sample Space S: possible outcomes in
rolling a six-sided die
Event E: odd numbered outcomes
Event F: even numbered outcomes
Example 2
Draw a Venn diagram to illustrate the following
probability problem: what is the probability of
getting a 5 on two consecutive rolls of the dice?
S
E
4 1
2 6 5
3
F 1
2 4
63
Tree Diagrams
Tree Diagram makes the enumeration of possible
outcomes easier to see and determine
N
N
Y
Y
Event 1
Event 2
N
N
Y
Y
N
Y
Event 3
N
Y
N
Y
HTT
HTH
HHT
HHH
Outcomes
TTT
TTH
THT
THH
Running the tree out details an individual outcome
Example 3
Given a survey with 4 “yes or no” type questions, list
all possible outcomes using a tree diagram. Divide
them into events (number of yes answers) regardless
of order.
Example 3 cont
N
N
Y
Y
Y
N
Y
Q1
Q2
Q3
N
N
Y
N
Y
N
Y
N
Y
N
Y
N
Y
N
Y
Q4
N
Y
N
Y
N
Y
N
Y
YNNN
YNNY
YNYN
YNYY
YYNN
YYNY
YYYN
YYYY
Outcomes
NNNN
NNNY
NNYN
NNYY
NYNN
NYNY
NYYN
NYYY
Example 3 cont
Outcomes
YNNN
YNNY
YNYN
YNYY
YYNN
YYNY
YYYN
YYYY
NNNN
NNNY
NNYN
NNYY
NYNN
NYNY
NYYN
NYYY
1
2
2
3
2
3
3
4
0
1
1
2
1
2
2
3
Number of Yes’s
0 1 2 3 4
1
4
6 4
1
Multiplication Rule
If you can do one task in n number of ways and
a second task in m number of ways, then both
tasks can be done in n  m number of ways.
Example 4
How many different dinner combinations can
we have if you have a choice of 3 appetizers, 2
salads, 4 entrees, and 5 deserts?
3  2  4  5 = 120 different combinations
Replacement
• With replacement maintains the original
probability
– Draw a card and replace it and then draw another
– What are your odds of drawing two hearts?
• Without replacement changes the original
probability
–
–
–
–
Draw two cards
What are you odds of drawing two hearts
How have the odds changed?
Events are now dependent
Example 5
From our previous slide:
• With Replacement:
(13/52) (13/52) = 1/16 = 0.0625
• Without Replacement
(13/52) (12/51)
= 0.0588
Summary and Homework
• Summary
– Probability is the proportion of times an event occurs in
many repeated trials
– Probability model consist of the entire space of outcomes
and associated probabilities
– Sample space is the set of all possible outcomes
– Events are subsets of outcomes in the sample space
– Tree diagram helps show all possible outcomes
– Multiplication principle enumerates possible outcomes
– Sample with replacement keeps original probability
– Sample without replacement changes original probability
• Homework
– Day One: pg 397 6-22, 24, 25, 29, 34, 36