**Unit 8: Probability**

Probability Notation and Sample Space

Probability notation: P(event) = Probability of the event occurring.

ex. There is a 30% chance of rain so:

P(rain) = 30/100 =0.30

ex. What's is the probability of rolling a 2 on a six-side dice?

P(rolling a 2) = 1/6

Independent, Mutually Exclusive, And, Or, NOT

Probability Using Data

Mr. Branch conducted a survey of the KHS Staff's favorite morning beverage.

**Probability is the extent to which an event is likely to occur, measured by the ratio of the favorable cases to the whole number of cases possible.**

P(event) =

Number of successful outcomes

Number of possible outcomes

Sample Space: A list, or table containing all possible outcomes.

Ex. The sample space for a 6 sided dice is 1,2,3,4,5,6.

Ex. 1 Use the table below to make a sample space for rolling two dice, one red and one green.

P(1,2)=

P(4,4)=

P(3,2)=

P(6,1)=

P(rolling at least one even)= P(sum is 6)=

P(roll at least one 4)= P(sum is 11)=

P(sum is 7)= P(difference is 3)= P(doubles)=

P(Doubles and sum is 8)= P(doubles or sum to 8)=

Ex 2. Sample Space and Probability for flipping 3 coins

P(HHH)=

P(TTT)=

P(HTH)=

P(TTH)=

P(1st flip H)=

P(2nd Flip T)=

P(1st & 2nd T)=

P(1st or 3rd H)=

P(One T) =

P(two H)=

**A**

**B**

**C**

**D**

Independent Events

Mutually Exclusive Events

Two events that have no effect on each other.

-Knowing one event occurs does not change the probability of the other.

Ex. Eating Pizza for lunch and if it rains.

Events that cannot occur at the same time or on the same outcome

ex. Drawing a King and a Ace on one draw.

Snowing and being 100 on the same day in GR.

And, Or, Not Statements

Warm up:

Imagine you are in the hot lunch line at school when you hear the lunch lady say the following statements:

Statement 1: "You need to pick a vegetable or a fruit."

Statement 2: "You need to pick a vegetable and a fruit"

Statement 3: "You may not take a fruit."

Not Statements:

* P(not A) means the probability of any event except A.

P(not A) = 1 - P(A)

*P(rain)=30% what is the P(it doesn't rain)?

P(not rolling a 3)= P(not spinning a B)=

P(not rolling an even)= P(not"rolling a 2 and spinning a B")=

P(not rolling a 5) = P(not spinning a Letter)=

And Statements:

In an "and" statement both events need to occur.

P(A and B) = P(A)*P(B)

Ex. 4 Use the table to the right to determine the probabilities below.

P(roll a 3 & spin a B)=

P(roll a 4 & spin a D)=

P(not roll a 2 & spin a C)=

P(spin an A on consecutive spins)=

P(roll an even & not spin a D)=

P(not roll a 1 & not spin a D)=

Or Statements:

In an "or" statement one or both of the events need to occur.

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

Ex. 5 Use the table to the right to determine the probabilities below.

P(roll a 2 or spin an A)=

P(roll a 3 or spin a C)=

P(not roll a 4 or spin a B)=

P(roll a 1 or not spin D)=

P(roll a 3 or roll a 4)=

P(spin an A or spin a B)=

*P(A and B) = the probability that events A and B both occur.

Multiplication Rule

Addition Rule

P(A or B)= Probability event A or event B or both occur.

*Why do we need to subtract P(A and B)?

And, or statements and Mutually Exclusive events

Give an example of mutually exclusive events from the situation of the four sided dice and the four section spinner. Call them event A and B.

Calculate the probability and explain what is means for:

P(A and B)=

P(A or B) =

Finish the sentence: For mutually exclusive events...

P(female)= P(female & Juice)=

P(male or water)= P(female or Coffee)=

P(coffee)= P(male & coffee)=

P(pop or juice)= P(water or female)=

P(juice and pop)= P(male & female)=

P(event)=-----------

event total

total data

Using the multiplication rule to determine if events are independent

*The multiplication rule only works for independent events. So two events are independent if:

P(X and Y) = P(X) * P(Y).

Are being female and liking juice independent events?

P(female):

P(juice):

P(female & juice)=

p(female)*p(juice)=

Multiplication rule:

Table

Use the multiplication rule to determine if the following events are independent events:

Ex 6: Being male and liking coffee

Ex 7: Being female and liking pop

Ex 8: Mr. Branch is having 9 friends over to watch the Michigan Game, how many of Mr. Branch's guests will choose juice as their favorite drink?

Are you comfortable making a prediction from this data? Explain.

Conditional Probability

The probability one event occurs given another event.

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

P(A & B)

P(B)

* The probability event A occurs given that event B has already occurred.

Ex 9: Explain what each conditional probability means in words.

*P(Juice|Female) = The probability of picking someone who likes juice out of all the females.

*P(Female|Juice) = The probability of picking someone who is female out of all the juice lovers.

P(Juice | Female) = ----------- = 5/12

5/30

12/30

P(Female | Juice) = ----------- = 5/8

5/30

8/30

A. P(water | Male) =

B. P(Pop | Female)=

C. P(Male | Coffee)=

D. P(Female| Water)=

E. P(Coffee | Female)=

F. P(Pop | Juice)=

G. P(Male| Water)=

A. P(water | Male) =

B. P(Pop | Female)=

C. P(Male | Coffee)=

D. P(Female| Water)=

E. P(Coffee | Female)=

F. P(Pop | Juice)=

G. P(Male| Water)=

Ex. 10 Calculate the conditional probability for each below.

Calculating Probability with Percentages (%)

* Remember percentages are numbers out of 100.

So 55% chance of rain means

P(rain) = 55/100 = 0.55

Consecutive probability and the Area Model

Create a probability distribution for drawing the specific type or card.

Warm- up

Create a probability table for the spinner below.

P(club)= P(K)= P(3 and Clubs)= P(red)=

P(A or Heart)= P(not J) = P(blue or Green)= P(not black)=

Consecutive Probability

*Finding the probability of events happening one after another.

*A and B are consecutive events if

A and B

happen one after another.

*The probability of event A and event B occurring consecutively is

P(A)*P(B)

8-section spinner

ExA. What is the probability of

spinning consecutive green sectors?

ExB. What is the probability it takes 4 spins to spin a black sector?

A deck of cards

ExC. What is the probability of drawing consecutive 2, if you return the card to the deck after the first draw.*

ExD. What is the probability it takes 5 draws to draw a black Jack?*

*How would C and D Change if we didn't put the cards back?

Ex. 11 Determine each probability below

A. spinning 3 blue sectors in a row.

B. spinning 4 red sectors is a row.

C. taking 4 spins to get a green sector.

Consecutive Probability

Ex. 12 Determine each probability below

(assume you return the cards to the deck)

A. Drawing a diamond on 3 consecutive draws.

B. Drawing a face card 5 draws in a row.

C. Taking 3 draws to pick a red 5.

Ex. 13 Determine each probability for example 12 if the cards are not returned to the deck.

Ex. 14 Miguel Cabrera has a batting average of .325 and Victor Martinez has a batting average of .335. What is the probability that...

A. they get a hit in consecutive at bats?

B. Miguel gets a hit and Victor does not?

C. What is the probability it takes Miguel 3 at bats before getting a hit?

Area Models:

A recent study has shown that 3/5 of people put ketchup on the side of their fries. Draw an area model to represent this statistic.

The same study has shown that 1/3 of all people do not use ketchup on their fries , draw an area model to represent this statistic.

Construct an area model that shows the probability that you use ketchup and put it on the side of your fries.