Probability Rules!!!

Section 5.2

Reference Text:

The Practice of Statistics, Fourth Edition.

Starnes, Yates, Moore

Objectives1. Probability Models

– Sample Space -Probability Model - Event

2. Basic Probability Rules– Compliment “not A” -Mutually exclusive (disjoint)

3. Addition Rule– P(A or B) =P(A) + P(B)

4. Two way tables

5. Venn Diagrams– Intersections and Unions.

Probability ModelsSome lingo to get down

• Toss a coin. What are the possible outcomes?• A: Heads or tails! This is what's know as our

sample space• Roll a regular 6-sided die. What are the possible

outcomes?• ____ ____ _____ _____ ____ ____• Probability model: some chance process that

consists of two parts: Sample space S, and probability for each one…. Lets look at rolling two 6-sided die! What are the outcomes?

Events

• With the two dice rolled, we could find any collection of outcomes and their probabilities.

• This is what’s known as an Event- any collection of outcomes from some chance process. Events are assigned capital letters such as A,B,C

• P(A) where A= sum of 5.

Compliment

Lets try some more!

• A = sum of 5

• B = sum is not 5

• C = sum is 6

• Find the probability of the following:

P(A)= P(B)=

P(C)= P(A or C)=

- Notice how P(A) +P(B) = 1

Basic Rules of Probability

Mutually Exclusive(disjoint)

• Two events are mutually exclusive if they have no outcomes in common and so can never occur together.

• Example: if one event occurs in 40% of all trials, and a different event occurs in 25% of the trials, and the two can never occur together, then one or the other occurs on 65% of the trials.

40% + 25% = 65%

Check for Understanding• Chose an American adult at Random. Define two events:

A = the person has a cholesterol level of 240 milligrams per deciliter of blood (mg/dl) or above. (High cholesterol)

B= The person has a cholesterol level of 200 to 239 (borderline high cholesterol)

• According to the American Heart Association:

P(A) = 0.16 and P(B) = 0.29

1. Explain why events A and B are mutually exclusive.

2. Say in plain language what the event “A or B” is. What is P(A or B)?

3. If C is the event that a person chosen has normal cholesterol (below 200 mg/dl) what's P(C)

Two way Tables• Students in college stats class wanted to find out

how common it is for young adults to have their ears pierced. They recorded data on two variables- gender and whether the student had a pierced ear – for all 178 people in class. The two way table below displays the data.

Pierced ears?

Gender Yes No Total

Male 19 71 90

Female 84 4 88

Total 103 75 178

A= male B= pierced ears• Suppose we chose a student from the class at

random. Find the probability that the student• (a) has pierced ears • (b) is a male with pierced ears • (c) is male or has pierced ears

Venn Diagrams

Event AEvent A P(A) = 90/178P(A) = 90/178

Event BEvent B P(B) = 103/178P(B) = 103/178

P(A and B) = 19/178P(A and B) = 19/178

Venn DiagramGeneral Addition Rule

• The Venn Diagram suggests to fix this “double counting”

• P(A or B) = P(A) + P(B) – P(A and B)

= 90/178 + 103/178 – 19/178

= 174/178

Intersection and Unions

• If we are talking about “A and B” then we can also call this the intersection of A and B. The corresponding notation is A П B

• If we are talking about “A or B” then we can also call this the Union of A and B. The corresponding notation is A U B

Venn Diagram: Intersection and Unions

Objectives1. Probability Models

– Sample Space -Probability Model - Event

2. Basic Probability Rules– Compliment “not A” -Mutually exclusive (disjoint)

3. Addition Rule– P(A or B) =P(A) + P(B)

4. Two way tables

5. Venn Diagrams– Intersections and Unions.

Test Results!

• Grade: Amount: Marginal %

• ……A......……....1.……….5%• …….B…………...9……...47% 68% Passed

• …….C…………..3..……...16%

• …….D…………..5.……...26%• …….F…..............1………..5% 31% Failed

• Mean: 79% Max: 92% Min: 58% No Outliers

Tracking AP Stats

• 2014-2015 (WHS)• Ch. 1 Test Ch. 2 Test Ch. 3 Test Ch. 4 Test• A 5 A 5 A3 A1• B5 B 6 B5 B9• C6 C 4 C6 C3• D2 D 1 D2 D5• F1 F 2 F2 F1

Homework

Worksheet