harrisonmathahs.weebly.comharrisonmathahs.weebly.com/.../spring2015unit8notes.docx · web...

Analytic Geometry Unit 8: Probability NotesName: _______________________________________________________________ Block: _______________

Unit 8: ProbabilityAfter completion of this unit, you will be able to…

Use set notation to represent a set of events mathematically Use the addition rule for two events or their probabilities Read and interpret a two way frequency table Determine conditional probabilities given sufficient information Use the formula for conditional probability Use the formula for independent events Determine whether two events are independent

Timeline for Unit 8Monday Tuesday Wednesday Thursday Friday

30Day 1

Set Notation & Venn Diagrams

1Day 2

Mutually Exclusive and Overlapping

4Day 3

Conditional Probability

5Day 4

Independent & Dependent

Events

6Day 5 Test

Analytic Geometry Unit 8: Probability Notes

Reference DiagramsDeck of Cards

Analytic Geometry Unit 8: Probability NotesAddition Table (Dice) Multiplication Table (Dice)

Day 1 - Probability Terminology, Notation, and Venn Diagrams

Analytic Geometry Unit 8: Probability NotesProbability

A number from 0 to 1 As a percent from ________ to __________ Indicates how likely an ________________will occur

In probability, a sample space is the set of all possible outcomes. Any subset from the sample space is an event.

Example: a. Sample Space: all the playing cards in a 52 card deck b. Event: drawing a queen of hearts Event: drawing a club Event: _________________

A single event is an event that describes a single outcome.Example:a. Flipping a coin and landing on headsb. Rolling a 3 with a die

A compound event combines two or more events using the word and or the word or.Example:a. Rolling a die two timesb. Flipping a coin four times.

The intersection of two or more events is all the outcomes shared by both events and is denoted with the word “and” or the symbol .

The union of two more events is all the possible outcomes for either events and is denoted with the word “or” or the symbol .

The complement of an event is the set of outcomes in the sample space that are not included in the outcomes of the event and is denoted with the word “not” or with the ‘ symbol.

We will use Venn Diagrams to help us visualize the probabilities that we discuss for the unit.

Example 1: Using results from our class, create a Venn Diagram and find the probabilities listed below:


Let A = Students taking a Science class

Let B = Students taking an English class

a. P(A) b. P(A)’c. P(B)

d. P(B)’

e. P(A B) f. P(A B)’ g. P(A B) h. P(A B)’

Example 2: We randomly selected 100 juniors enrolled in Pre-Calculus at AHS. Of those 100 students, 55 are taking AP Calculus, 35 are taking AP Stats, and 10 are taking both courses. Construct a Venn Diagram and answer the questions:

Example 3: Each member of a sports club plays at least one of soccer, rugby, or tennis. The following is known: 43 members play tennis, 11 play tennis and rugby, 7 play tennis and soccer, 6 play soccer and rugby, 84 play rugby or tennis, 68 play soccer or rugby, and 4 play all three sports. Create a Venn Diagram to represent the members and the sports they play.

1. How many members are there total?

2. Find the following probabilities:

P(R T) P(T)’

P(S R) P(S T)’

P(T R’) P(R S’)

2. Find the following probabilities:a. P(Calc or Stats)

b. P(Not Calc or Not Stats)

c. P(Not Calc)

d. P(Not Stats and Calc)

Analytic Geometry Unit 8: Probability NotesDay 2 - Mutually Exclusive Events & Overlapping Events

If two or more events cannot occur at the same time, they are considered mutually exclusive. This means they have no common outcomesExample: a. Rolling a 1 and rolling at 2 on the same roll with a die cannot occur at the same time.b. Flipping a coin and getting heads and tails cannot occur at the same time.

Example 1: A drink company applies one label to each bottle cap: “free drink,” “free meal,” or

“try again.” A bottle cap has a probability of being labeled “free drink” and a probability of being labeled a “free meal.”

a. Explain why the events “free drink” and “free meal” are mutually exclusive.

b. What is the probability that a bottle cap is labeled “free drink” or “free meal.”

Example 2: The table below describes the probability of the following stores being classified as a girl’s favorite department store. Using the table below, find the following probabilities:

a. P(Macy’s or Nordstrom):

b. P(not JC Penney’s):

Example 3: Using the table below that represents the sum of rolling two dice,

If two events are considered mutually exclusive, the probability that the events will occur is found by adding the individual probabilities of the events:

P(A or B) = P(A) + P(B)


a. What is the probability of rolling a sum of 11?

b. What is the probability that your sum will be a 4 or 5?

Overlapping Events

If two or more events have at least one outcome in common, they are called overlapping events.

Example: a. Rolling a prime number on a die or rolling an even number on a die would have an overlapping event of rolling a 2 (2 is prime and even)

Prime Numbers {2, 3, 5}Even Numbers {2, 4, 6}

Explanation: Therefore I have 6 different outcomes out of 36 possible outcomes, but I don’t want to include “2” twice since it is the same number, therefore, I am going to take the probability of rolling a prime + probability of rolling an even – probability of rolling a 2

P(rolling a prime or even)=

Example 1: Find the probability:a. Rolling a 5 or an odd number on a die:

b. Rolling an even sum or a sum greater than 10?

If two events are considered overlapping events, the probability that the events will occur is found by adding the individual probabilities of the events minus the probability of both occurring:

P(A or B) = P(A) + P(B) – P(A B)


Example 2: Using the deck of cards below, find the following probabilities:a. Drawing a king or heart:

b. Drawing a red card or a face card:

Example 3: Using the Venn diagram below, let B = band members and A = club members at Lewis High School.

a. P(B) c. P(A B)

b. P(A B) d. Find P(A B)’

Example 4: Using the table below, find the probability of picking a female or a person from Florida.


Day 3 - Conditional Probability

Conditional Probability: Contains a condition that may limit the sample space for an event. You can write a conditional probability using the notation:

P(B|A)=P(B given A) The formula for conditional probability is:

1. The table shows the results of a class survey. Find P(own a pet | female).

2. The table shows the results of a class survey. Find P(wash the dishes | male)

3. Using the data in the table, find the probability that a sample of not recycled waste was plastic. P(plastic | non-recycled)

4. Researchers asked people who exercise regularly whether they jog or walk.Fifty-eight percent of the respondents were male. Twenty percent of all respondents were males who said they jog. Find the probability that a randomly selected person jogs given they are male.

5. A bag contains blue and yellow marbles. Two marbles are drawn without replacement. The probability of selecting a blue marble and then a yellow marble is .37 and the probability of selecting a yellow marble on the second draw, if the first marble drawn was blue is .67. What is the probability of selecting a blue marble?

Do you own a pet?

Yes NoFemale 8 6

Male 5 7

Did you wash the dishes last night?

Yes No

Female 7 6Male 7 8

Material Recycled Not RecycledPaper 34.9 48.9Metal 6.5 10.1Glass 2.9 9.1Plastic 1.1 20.4Other 15.3 67.8

Analytic Geometry Unit 7: Probability NotesDay 4 – Independent and Dependent Events

Independent Events:Two events A and B, are independent if the fact that A occurs does not affect the probability of B occurring.

Examples:

-Landing on heads from two different coins -Rolling a 4 on a die, then rolling a 3 on a second roll of the die

Independent Event Formula: Probability of A and B occurring:

1. A coin is tossed and a 6-sided die is rolled. Find the probability of landing on the head side of the coin and rolling a 3 on the die.

2. A card is chosen at random from a deck of 52 cards. It is then replaced and a second card is chosen. What is the probability of choosing a jack and an eight?

3. A jar contains three red, five green, two blue and six yellow marbles. A marble is chosen at random from the jar. After replacing it, a second marble is chosen. What is the probability of choosing a green and a yellow marble?

4. A school survey found that 9 out of 10 students like pizza. If three students are chosen at random with replacement, what is the probability that all three students like pizza?

Analytic Geometry Unit 7: Probability NotesDependent Events:Two events A and B, are dependent if the fact that A occurs affects the probability of B occurring.

Examples:

-Picking a blue marble out of a bag and then picking another blue marble without putting the 1st one back in the bag.

Dependent Event Formula: Probability of A and B occurring:

5. A jar contains three red, five green, two blue and six yellow marbles. A marble is chosen at random from the jar. A second marble is chosen without replacing the first one. What is the probability of choosing a green and a yellow marble?

6. An aquarium contains 6 male goldfish and 4 female goldfish. You randomly select a fish from the tank, do not replace it, and then randomly select a second fish. What is the probability that both fish are male?

7. A random sample of parts coming off a machine is done by an inspector. He found that 5 out of 100 parts are bad on average. If he were to do a new sample, what is the probability that he picks a bad part and then, picks another bad part if he doesn’t replace the first?

8. Tim’s goal is to run one marathon in each of the fifty states. To determine the order in which he will run the marathons, he will write the name of each state on a slip of paper and place the slips of paper in a bowl. He will draw the names of the states one at a time from the bowl until all the slips of paper have been drawn.

If there are 26 states east of the Mississippi River and 24 states west of the Mississippi River, what is the probability that the third state drawn will be east of the Mississippi, GIVEN THAT the first one drawn was east and the second one drawn was west of the Mississippi?


How to Determine if Two Events are Independent: Two events are independent if the following are true:

P(A|B) = P(A) P(B|A) = P(B)

You must prove one of the above conditions to prove two events are independent

9. Let event G = taking a math class. Let event H = taking a science class. Then, G AND H = taking a math class and a science class. Suppose P(G) = 0.6, P(H) = 0.5, and P(G AND H) = 0.3. Are G and H independent?

10. In a particular college class, 60% of the students are female. 50% of all students in the class have long hair. 45% of the students are female and have long hair. Let F be the event that the student is female. Let L be the event that the student has long hair. One student is picked randomly. Are the events of being female and having long hair independent?

harrisonmathahs.weebly.comharrisonmathahs.weebly.com/.../spring2015unit8notes.docx · web...

Documents