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 Heads Tails

1

51

251

0 1

18 33

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

E F

13

5 2 46

Sample Space S: possible outcomes in rolling a six-sided die

Event E: odd numbered outcomesEvent F: even numbered outcomes

S

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?

E F1

35 2 4

6

S

246

1

3

Tree Diagrams

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

Running the tree out details an individual outcome

Event 1

Y

N

Event 2

Y

N

Y

N

Event 3

Y

N

Y

N

N

Y

Y

N

HTT

HTH

HHT

HHH

TTT

TTH

THT

THH

Outcomes

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

Q 1

Y

Q 2

Y

N

Q 3

Y

N

Y

N

YNNNYNNYYNYNYNYYYYNNYYNYYYYNYYYY

OutcomesQ 4

NYNYNYNY

N

Y

NY

N

Y

N

NYNYNYNY

NNNNNNNYNNYNNNYYNYNNNYNYNYYNNYYY

Example 3 contYNNN 1YNNY 2YNYN 2YNYY 3YYNN 2YYNY 3YYYN 3YYYY 4NNNN 0NNNY 1NNYN 1NNYY 2NYNN 1NYNY 2NYYN 2NYYY 3

OutcomesNumber 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