Lesson 6

1
Lesson 6 2a

• Probability Models

2
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.

3
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.

4
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.

5
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.

6
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

7
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)

8
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.

9
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.

10
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 Heads Tails
1
51
251

• 0 1
• 33
• 117 134

11
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
F
5
2
1
4
3
6
Sample Space S possible outcomes in rolling a
six-sided die Event E odd numbered
outcomes Event F even numbered outcomes
12
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?

13
Tree Diagrams

• Tree Diagram makes the enumeration of possible
  outcomes easier to see and determine

N
HTT HTH HHT HHH
N
Y
Y
N
Y
Y
Event 1
Event 2
Event 3
Outcomes
N
TTT TTH THT THH
N
Y
N
N
Y
Y
Running the tree out details an individual outcome
14
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.

15
Example 3 cont
YNNN YNNY YNYN YNYY YYNN YYNY YYYN YYYY
N
N
Y
N
N
Y
Y
Y
N
N
Y
Y
N
Y
Y
Q 1
Q 2
Q 3
Outcomes
Q 4
NNNN NNNY NNYN NNYY NYNN NYNY NYYN NYYY
N
N
Y
N
N
Y
Y
N
N
N
Y
Y
N
Y
Y
16
Example 3 cont
YNNN 1 YNNY 2 YNYN 2 YNYY 3 YYNN 2 YYNY 3 YY
YN 3 YYYY 4 NNNN 0 NNNY 1 NNYN 1 NNYY 2 NYNN
1 NYNY 2 NYYN 2 NYYY 3
Number of Yess Number of Yess Number of Yess Number of Yess Number of Yess
0 1 2 3 4
1 4 6 4 1
Outcomes
17
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.

18
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
19
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

20
Example 5

• From our previous slide
• With Replacement (13/52) (13/52) 1/16
  0.0625
• Without Replacement (13/52) (12/51)
  0.0588

21
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