MIME-Version: 1.0 Content-Type: multipart/related; boundary="----=_NextPart_01CA8E18.5C3DE540" This document is a Single File Web Page, also known as a Web Archive file. If you are seeing this message, your browser or editor doesn't support Web Archive files. Please download a browser that supports Web Archive, such as Microsoft Internet Explorer. ------=_NextPart_01CA8E18.5C3DE540 Content-Location: file:///C:/45195D13/chapter_12_skeleton_notes.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Chapter 12 Skeleton Notes

Topics in Mathematics

Chapter 12 Homework Assignments

 

 

Section 12.1        10-14 all, 15-29 odds

 

Section 12.2        1, 2, 3, 5, 7, 9, 11, 12, 17-71 odds

 

Section 12.3        1-7 odds, 11-53 odds

 

Section 12.5        1, 2, 4, 5, 7-19 odds

 

Section 12.1:&n= bsp; Probability

 

Experiment: 

 

 

Outcomes: 

 

 

Event: 

 

 

Theoretical Probability (Mathematical): 

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&nbs= p;

Empirical Probability (Experimental): 

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&nbs= p;

&nbs= p;

Example:  Suppose we flip a fair coin.

&nbs= p;

What is = the percent chance of flipping a head? &nb= sp;         &= nbsp;  

&nb= sp;

Flip a c= oin 10 times and record the results.

&nb= sp;

Fl= ip 1

Fl= ip 2

Fl= ip 3

Fl= ip 4

Fl= ip 5

Fl= ip 6

Fl= ip 7

Fl= ip 8

Fl= ip 9

Fl= ip 10

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&nbs= p;

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&nbs= p;

&nbs= p;

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&nbs= p;

&nb= sp;

What per= cent heads did you flip ()?         

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Will the Theoretical and Empirical probabilities always match?            =   

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&nbs= p;

Law of Large Numbers:<= /span> 

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&nbs= p;

&nbs= p;

&nbs= p;

Determining Empirical Probability

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Probability of an Event:  P(E) =3D= =

 

&nbs= p;

Example:  40 people were surveyed on their favorite flavor of ice cream.  Find the empirical probability that a person liked each of the flavors.

&nbs= p;

Flavor

Survey Results<= /o:p>

Computation

Empirical Probabilit= y

Vanilla

18=

 

 

Chocolate=

11=

 

 

Strawberry

4<= o:p>

 

 

Other

7<= o:p>

 

 

&nbs= p;

&nbs= p;

Example:  55 people were surveyed on the num= ber of siblings (brothers and/or sisters) in their family.  Find the empirical probability tha= t a person had the following number of siblings.

&nbs= p;

# of Siblings

Survey Results<= /o:p>

Computation

Empirical Probabilit= y

0<= /b>

8

 

 

1<= /b>

13

 

 

2<= /b>

17

 

 

3<= /b>

10

 

 

4<= /b>

4

 

 

Other

3

 

 

&nbs= p;

 

 

&nbs= p;

&nbs= p;

&nbs= p;

&nbs= p;

&nbs= p;

&nbs= p;

&nbs= p;

Section 12.2:  Theoretical Probability=

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Equally Likely Outcome= s: 

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&nbs= p;

&nbs= p;

Probability in an Expe= riment with Equally Likely Outcomes

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P(E) =3D= =

&nbs= p;

&nbs= p;

&nbs= p;

Exam= ple:  A card is drawn from a standard deck of cards.  Find = the probability of drawing:

 

a)         = an ace      &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;    

 

 

b)         = a face card     &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;          

 

 

c)         = an eleven      &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;

 

 

d)         = a heart      &n= bsp;            = ;            &n= bsp;            = ;            &n= bsp;    

 

 

e)         = anything but a heart     &n= bsp;            = ;            &n= bsp;           

 

 

f)        &= nbsp; a card shaped like a rectangle  &nb= sp;            =             <= /p>

 

 

g)         = a 4 or a spade     &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;      

 

 

h)         = a 4 and a spade     =             &nb= sp;            =             &nb= sp;    

 

&nbs= p;

Properties of Probabil= ity

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1)     

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&nb= sp;

&nb= sp;

2)     

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&nb= sp;

&nb= sp;

3)     

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&nb= sp;

&nb= sp;

4)     

 

 

 

5)     

 

 

 

Example:  A fair coin is flipped = two times.

 <= /o:p>

a)         = Determine all possible outcomes.   &nb= sp;            =             &nb= sp;            =      

 <= /o:p>

 <= /o:p>

b)         = Find the probability of flipping two heads in a row.                &= nbsp;  

 <= /o:p>

 <= /o:p>

c)         = Find the probability of not flipping two heads in a row.  

 

Example:  In the daily number, three ping po= ng balls are selected from three different machines.  Each machine contains balls number= ed 0-9.  Find the probability of:

=  

a)         = hitting your favorite number.   &nbs= p;            &= nbsp;           &nbs= p;

=  

=  

b)         = not hitting your favorite number.  &n= bsp;            = ;        

 

 

 

Section 12.3:  Odds: 

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&nbs= p;

 

 

 

Odds Against (Most Common): 

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Odds Against =3D<= /o:p>

 

 

 

Example:  Determine the odds against drawing= an Ace from a deck of cards.

 

P(success) =3D &nb= sp;            =             &nb= sp;          P(failure) =3D<= /b>

 

 

Odds Against =3D

 

 

 

Note: 

&nbs= p;

&nbs= p;

&nbs= p;

Example:  Determine the odds against rolling= a five or six on a die.

&nbs= p;

P(success) =3D &nb= sp;            =             &nb= sp;          P(failure) =3D<= /b>

 

 

Odds Against =3D

 

 

&nbs= p;

Odds in Favor (Less Co= mmon): 

&nbs= p;

&nbs= p;

&nbs= p;

Odds in Favor =3D=

 

Example:  A fair coin is flipped = two times.  Find the odds in favor= of flipping two heads in a row.

&nbs= p;

P(success) =3D &nb= sp;            =             &nb= sp;          P(failure) =3D<= /b>

 

 

Odds in Favor =3D

 

 

 

Example:  Two dice are rolled.  Find the odds in favor of rolling “snake eyes” (two ones).

&nbs= p;

P(success) =3D &nb= sp;            =             &nb= sp;          P(failure) =3D<= /b>

 

 

Odds in Favor =3D

&nbs= p;

&nbs= p;

&nbs= p;

Determining Probabilit= ies Given Odds

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1)     

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&nb= sp;

2)     

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&nb= sp;

3)     

&nb= sp;

&nb= sp;

Note)=

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&nbs= p;

Example:  The odds against winning the daily number are 999/1.&nb= sp; Find the probability of winning and the probability of losing.

 

1)            =             &nb= sp;          2)            =             &nb= sp;          3)     

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Example:  The odds against the Pirates winni= ng the World Series are 25,000/1.  Find the probability of the Pirates winning and the probability of the Pirates not winning.

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1)            =             &nb= sp;          2)            =             &nb= sp;          3)     

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Section 12.5  Tree Diagrams

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Counting Principle: 

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&nbs= p;

&nbs= p;

Example:  We roll two dice.  Find the number of possible outcom= es.

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&nbs= p;

&nbs= p;

Example:  We flip three coins.  Find the number of possible outcom= es.

&nbs= p;

&nbs= p;

&nbs= p;

Sample Space: 

&nbs= p;

&nbs= p;

Tree Diagram: 

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&nbs= p;

&nbs= p;

Example:  Produce the sample space for flipp= ing three coins using a tree diagram.

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&nbs= p;

&nbs= p;

&nbs= p;

&nbs= p;

&nbs= p;

&nbs= p;

&nbs= p;

Example:  A bag contains four chips colored = blue, green, red, and yellow.  One c= hip is selected and then, without replacement, a second chip is selected.  Produce the sample space for this experiment using a tree diagram.

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&nbs= p;

&nbs= p;

Example:  Two spins are given on a spinner t= hat has 4 evenly-distributed colors on it, red, blue, yellow, and green.

A)        How many total outcomes?    = ;   Use the counting principle and a tree diagram.

B)        What is the probability of RedRed?

C)        What is the probability of at least one of the spins being Red?

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&nbs= p;

&nbs= p;

&nbs= p;

&nbs= p;

&nbs= p;

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Example:  Betty is going to paint her living= room, dining room and kitchen.  She’s narrowed the colors as follows:

Living Room:  purple, orange, or gree= n

Dining Room:  green, red, or yellow

Kitchen:=   blue or red

Each col= or is equally likely to be picked.

A)        How many total outcomes?    = ;   Use the counting principle and a tree diagram.

B)        What is the probability of a color scheme that includes green?

C)        What is the probability of a color scheme that doesn’t include red?

&nbs= p;

&nbs= p;

&nbs= p;

&nbs= p;

&nbs= p;

&nbs= p;

&nbs= p;

&nbs= p;

Example:  Jill has six socks in a drawer.  Two socks are blue, two are tan, a= nd two are white.  In the dark, she s= elects one sock, then without replacement, selects a second sock.

A)        How many total outcomes?    = ;   Use the counting principle and a tree diagram.

B)        What is the probability of selecting two tan socks?

C)        What is the probability selecting any “match”?

D)        What is the probability of not selecting a “match”?

&nbs= p;

&nbs= p;

&nbs= p;

&nbs= p;

&nbs= p;

&nbs= p;

&nbs= p;

&nbs= p;

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Chapter 12  =             &nb= sp;            =             &nb= sp;            =             &nb= sp;            =             &nb= sp;            =             &nb= sp; Page 1

Probability

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