9.5 Probabilities of Various Combinations

This section discusses another way to calculate probability of specific situations. Also, there are some new (easier) notations, and some calculator functions! If you want, you can call this section “Permutations VS Combinations,” as Mr. French has listed it for Friday’s quiz topics list. Being able to distinguish the combination from the permutation is pretty important, and probably will be for this next quiz!

Terminology:
1. Permutations: Looking @ the # of possible outcomes in a situation when order is important (as we learned from Nick yesterday).

2. Combinations: Looking @ the # of possible outcomes in a situation when order is not important.

A number of combinations = total number of permutations / # of permutations of each one combination

So, when we are asked on the quiz: according to their definitions, what is the conceptual difference between a permutation and a combination? We’ll nail it! ☺ Significance of order is what counts!!

Notation:
1. Permutations: nPr = n!/(n-r)!
2. Combinations: nCr = n!/(n-r)!r!
• n = total
• r = amount being chosen
• (n-r) = amount not being chosen, a.k.a. amount left

Examples:
Permutations: You’re hosting the after party for a school dance, say, last Friday’s Toga Dance. You’ve got a huge couch that can seat 10 people- no sitting on laps allowed, according to your parents, since heaping on more than maximum carrying capacity will dent the seat pillows beyond repair. Anyway, fifty people show up. They all want to sit on the couch. How many possible ways could you seat the guests?

n = total = 50
r = amount being chosen = 10
So…
50P10 = 50!/(50-10)! BUT WAIT!

Nick taught us yesterday how to do individual factorials by calculator…but we can also do the entire permutations with one function, so lets do that.

• Enter n, which is 50.
• Click MATH, then go to PRB (all the way to the right)
• Scroll down to & choose the nPr option
• Enter r, which is 10.

DANG! The answer is 37,276,043,020,000,000.

Combinations: Same situation, but this time, you want to know how many combinations you can make. The order of the people don’t matter, you just can’t make the same combination more than once.

n = total = 50
r = amount being chosen = 10
So…
50C10 = 50!/(50-10)!10! BUT WAIT! AGAIN!

Like the permutation calculator shortcut, there’s also a combination shortcut. The only thing you need to do differently is that when you scroll down in the PRB section, choose the nCr option.

The answer will be 10,272,278,170 – only.

• Keep in mind that there will be less combinations than permutations. Because order matters, several permutations could be different arrangement of the same combination.

Concept Examples:
Permutations: Write 8P3 as ratio of factorials. Write the answer in terms of the number of elements of the set & number of elements selected.
Answer: 8!/5!
• 8P3 = 8!/(8-3)! = 8!/5!

Combinations: Write 6C2 as a ratio of a factorial to a product of factorials.
Answer: 6!/4!2!
• 6C2 = 6!/(6-2)!2! = 6!/4!2!

Combinations in terms of Permutations: This time, write 6C2 as a ratio of permutations
Answer: 6P2 / 2P2
• Looking back to the Terminology section:
A number of combinations = total number of permutations/# of permutations of each one combination

So… 6C2 = 6P2 / 2P2

That’s it guys! If anybody needs any extra help, here’s a good, simple site:
http://www.themathpage.com/aPreCalc/permutations-combinations.htm

And for those of you that like really convoluted explanations with pretty pictures, here’s another good one, just in case (hey, additional sample problems!):
http://www.mathagonyaunt.co.uk/STATISTICS/ESP/Perms_combs.html

MADDIE’S NEXT! Ah… barely missed the weekend.

Prep Enthusiasm
I’d like to dedicate this personalization to one of our Prep geniuses: DR. PARKER! (Sorry, Mr. French. We all know of your genius, but you already have an entire interview in one of the calculus blogs!!) Dr. Parker is a crazy composer, as in CRAZY GOOD! Sometimes, the instrumental music dept. gets the opportunity to play some of his works for plays, choral & orchestral, etc. But what I really wanted to get out there was that he has a website with tons of recordings that you can download or listen to for free. It’s www.robertparkermusic.com. It’s really quite an impressive collection in terms of both quantity & quality.

Friday's Quiz Topics

Here’s a list of topics for Friday’s quiz:

Precalculus Quiz 9.1-5 Topics
Definition of terms
Probability – rolling a die (and/or events)
Calculating outcomes – and/or
Permutations vs. combinations
Calculating permutations and combinations
Interpretation of combinations
Variations of combinations and permutations – calculating outcomes and probabilities
Repeated elements in calculations

That’s it! The format’s the same as always – ½ Non-calculator, ½ Calculator. There are 9 questions on the Non-calculator portion and 14 questions on the Calculator section. I’ll be around after school on Wednesday, and hopefully accessible by email on Thursday. Mr. Frost should be able to answer any questions you have during class.

In any collection of data, the figure most obviously correct,
beyond all need of checking, is the mistake

Corollaries:
(1) Nobody whom you ask for help will see it.
(2) The first person who stops by, whose advice you really
don't want to hear, will see it immediately.

And on another note - look for the simple solution!

9-4: Permutations

This is Chapter 9 Section 4: Permutations (or as the book refers to it: "Probabilities of Various Permutations" -- which is a mouthful.

Definition: Permutation
A Permutation is a way to count the number of outcomes when order is
important.

The best way to explain this is through examples. Here are some examples of different times a permutation may help you:

Example 1
We want to form a line with three students to take a picture. How many ways can we do this, assuming there are twelve of us in the classroom:

12 11 10

There would be 1,320 different ways to take this picture. (12x11x10)

Example 2
What if there is a restriction? For example, I am short and want to be in the middle. Let's see how we can do this if we want to be able to take 5 photos:

11 10 1 9 8

The 1 is me in the middle because there is only one way to make this work and the rest is 11, 10, 9 and 8.

Factorials!
This is what makes our life easier. Let's say that we had twelve students and twelve different ways to take the photo... so it's...

12 11 10 9 8 7 6 5 4 3 2 1

The easiest way to write this is: 12!

Our calculator can help make this really easy:

1. Enter the number
2. Go to Math and then PRB (probability)
3. Scroll to the ! which stands is the option for a factorial.

By the way, the answer is 479, 001, 600

Here's a hint, fill the restrictions first and then continue with the factorial.

Example Problem

You are designing a new frozen yogurt shop and must decide the order for all the different toppings. You know that there are 21 choices and that "Strawberry" must be the first choice. Determine the possible amount of ways to organize your toppings.

Answer...

1 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

This is the breakdown of my choices. Note that there is only one option for the first topping (Strawberry). To solve, use 1 x 20! in your calculator which gives you...

2.43 x 10 ^ 18

Which is a lot of different ways to organize your store.

Extra Help:

http://mathforum.org/dr.math/faq/faq.comb.perm.html

Describes permutations as well as combinations-- so only look at the first part. It's a fairly good explanation.

Amira you're up next!

Soo... I would like to share with everyone my new favoritest thing ever. In fact, over the weekend I went and brought two large ones home which I'm eating right now.

I suggest that you read about this amazing non-fat all natural creation and then visit your nearest Pinkberry! (Mine's Studio City, but they will open in Old Town sometime this year) Don't get fooled and visit Roseberry in Glendale-- it's not the same. Ironically enough, Pinkberry is in many ways a copy of a huge Korean chain called Iceberry... but enough talking.

9.3 Two Counting Principles

9.3 Two Counting Principles

Independent Event:

The way one event occurs does not affect the way the other could occur.

Mutually Exclusive events:

The occurrence of one event excludes the possibility the other will occur.

n(A or B) = n(A) + n(B)

(the number of A or B is the number of A + the number of B)

Properties: Two Counting Principles:

Let A and B be two events that occur in sequence.

Then n(A and B) = n(A) * n(B)

Where n(BA) is the number of ways B can occur after A has occurred.

When A and B are independent = n(A) * n(B)

Overlapping Events:

n(A or B) = n(A)+ n(B) -n(A B)

(number of ways A can occur + number of ways B can occur – number of ways A and B overlap)

· means “and” on the intersection of A and B

· No common area means that A and B are mutually exclusive and “-n(A B)”

equals zero

Example Problem:

A salad menu consists of 4 toppings and 5 dressings. Find the number of different ways you could select..

a. a topping or a dressing

b. a topping and dressing

Work:

a. Mutually Exclusive events: n(A or B) = n(A) + n(B); n(dressing or topping) = n(4) + n(5) = 9

b. Two Counting Principles where they are both independent: n(A and B) = n(A) * n(B); n(dressing and topping) = n(dressing) * n(topping) = 20

Answer

a. 9

b. 20

Extra Help:

http://www.gomath.com/htdocs/lesson/probability_lesson1.htm

Reminder to NICK LOUI!!! You are up next!!!

I am about to go to a prom meeting so here is a fun little fact (sorry sophomores): PROM IS IN 54 DAYS!! Start thinking about dresses, dates, and plans!!!

Lesson 9-2 : Words Associated With Probability

Awesome...I'm so lucky to get the words associated with probaility.

Alright, now let's get to the lesson. It's really all definitions and not nearly as hard as some of the lessons we had beforehand.

In the book, they used a dice rolling experiment so i'll just go along with it...

Random Experiment - The act of doing something .
In the experiment, it would be the act of rolling the dice.

Trial - Each time you do something.
In the experiment, it would be each time you rolled the dice.

Simple Event or Outcome - What you get.
The results of the experiment.

Event - Set of outcomes

Sample Space - All of the possible outcomes.

Probability - Number of outcomes in the event / Number of outcomes in the sample space

Sample Problem
Given an unlimited amount of dimes, quarters, nickles, and pennies. Find how many possible ways there are to get 50 cents if you have to use a dime, quarter, and nickle at least once. Then find the probability that you will have to use 2 or more nickles.

Well first we can find the amount, that we have to use. So if we have to use at least one for all three that means that we must have at least 25+10+5. That equals 40 just so you all know.

Then we find how many different ways we can make 10 cents.
1) We can add 1 dime.
2) We can add 2 nickles.
3) We can add 10 pennies.
4) We can add 5 pennies and 1 nickle.

Now we know that there are 4 possible ways to create 50 cents while using a dime, quarter, and nickle at least once.
We see that the nickle is used at least once in all of them, so we just have to find the events where the nickle is used more than once, which is 2.
Now we put the number of outcomes in the event / number of outcomes in the sample space, which is 2/4 or 1/2.

Answer: 4 possible ways. 1/2 probability that we will use 2 or more nickles.

Wow...this website is basically everything i just typed up...with a lot of examples!
http://www.mathgoodies.com/lessons/vol6/intro_probability.html

Reminder!!! Katie you are our next blogMASTER. have fun. =]

Well people say laughter is the best medicine, and well YouTube has plenty of videos to make us all laugh!!! Hahahaha. Here are two really good ones.
http://youtube.com/watch?v=Z4Y4keqTV6w
http://youtube.com/watch?v=5P6UU6m3cqk

Thursday's Test Topics

Here’s a list of topics for Thursday’s test:

Precalculus Chapter 7 Test Topics:
General equations (no translations) for all the functions
Sketch graphs of all the functions
Determine equations from graphs
Logarithmic properties
Exponential to logarithm and logarithm to exponential
Properties of exponents
Patterns for all functions
Determine points given types of functions
Demonstrate patterns from data tables
Find equations from data tables
Concavity
Find additional points given a data table
Demonstrate pattern given equation
Determine x-values algebraically from equations given y-values
Logistic functions – description, soutions, point of inflection

That’s it! The format is as expected – ½ calculator, ½ non-calculator. I’ll be in my classroom on Wednesday after 3:30 and before school on Thursday. I’ll also be available online later on Wednesday evening, after the girl’s basketball game. If you have specific questions Wednesday night, email me!

See you in class!

At New York's Kennedy Airport today, an individual, later discovered to be a public school teacher, was arrested trying to board a flight while in possession of a ruler, a protractor, a set square, and a calculator. Attorney General John Ashcroft believes the man is a member of the notorious Al-Gebra movement. He is being charged with carrying weapons of math instruction.
Al-Gebra is a very fearsome cult, indeed.They desire average solutions by means and extremes, and sometimes go off on a tangent in a search of absolute value. They consist of quite shadowy figures, with names like "x" and "y", and, although they are frequently referred to as "unknowns", we know they really belong to a common denominator and are part of the axis of medieval with coordinates in every country. As the great Greek philanderer Isosceles used to say, there are 3 sides to every angle, and if God had wanted us to have better weapons of math instruction, He would have given us more fingers and toes.
Therefore, I'm extremely grateful that our government has given us a sine that it is intent on protracting us from these math-dogs who are so willing to disintegrate us with calculus disregard.
These statistic bastards love to inflict plane on every sphere of influence. Under the circumferences, it's time we differentiated their root, made our point, and drew the line. These weapons of math instruction have the potential to decimal everything in their math on a scalar never before seen unless we become exponents of a Higher Power and begin to appreciate the random facts of vertex.
As our Great Leader would say, "Read my ellipse". Here is one principle he is uncertainty of---though they continue to multiply, their days are numbered and sooner or later the hypotenuse will tighten around their necks.

7.6 Logistic functions, are cool.

Logistic Function 7-6



Logistic =exponential functions for restrained growth




These functions grow exponentially up to a critical point (x, c/2) then they level off. C is an asymptote for the graph. It is sometimes known as the carrying capacity.







The standard equation for a logistic graph is f(x) = C/ (1+ ab^(-x)) or f(x) = C/ (1+ae^(-bx))

thats all you need to know! heres an example problem...

Example- Bob wants to paint his 100 square foot wall blue. The amount of wall he paints as a function of time is logistic because he starts out slowly because he has never painted a wall before, he gets the hang of it and speeds up, then towards the end he slows down because he is tired and has to be careful.

x(hours)---- y(square feet painted)
1 -----------10
2 -----------25
3 -----------50
4 -----------80





Use the first and last points to find an equation for the function.

Answer -



aight i'm done...


For more, but not better, information. See http://en.wikipedia.org/wiki/Logistic_function.





Larry/Edward you're next!!?!?!??!

7-5 : Logarithimic Functions!!!

Hey folks...I am here...to teach you...all about...LOGARITHIMIC FUNCTIONSSS CUZ IM RICK JAMES silly goose. So...let's get started


We all know that a logarythm is the inverse of an eponential function. Therefore we can deduce (big word) that if given a set of data, and it is a logarhythm, then it will follow the multiply - add pattern. Observe...

x y

6 1
18 2
54 3
162 4
486 5
1358 6

the x-values are multiplied by a factor of 3, every time the y-values add 1.


Recap of what a log function is :
y =a + b log x
a graph of this would look like this (notice that it contains the point (1,0))

























Ok now a little bit more algebra stuff for all of you guys...*using the formula, y = a + b ln(x)*

1 = a + b ln(6)

2 = a + b ln(18)

subtract

-1 = b ln(6) - b ln(18)

take out b

-1 = b[ ln(6) - ln(18) ]

use quotient property

-1 = b[ ln(1/3) ]

divide both sides by ln(1/3)

b = -1/ln(1/3)

b = .910

ok now we solved for b, next is a

1 = a + .910 * ln(6)

isolate a

a = 1 - .910 * ln(6)

a = -.631

the answer is *drumroll please*

y = -.631 + .910 * ln(x)





Ok now I get the pleasure of showing you how to translate and dilate logarythms!!
in red is y = ln(x)
in blue is y = 2ln(x-1).........(notice that its asymptote is moved over to x=1, and that it does not taper off as quickly)






























Anything in the parenthesis with the x, affects the graph horizonatally, anything outside affects the graph vertically, just like always.





now this is a special case...this is the graph of log (x squared - 1)
the grey lines represent asymptotes, the red lines cross the x-axis at (-2, 0) and (2, 0)

My Problem

the starting logarhythm function is y = 5 + ln(2x), increase the horizontal dilation by a factor of 2 and the vertical dilation by a factor 9.

answer : y = 5 + 9 * ln(x)


OK that is lesson 7-5 in a nutshell
reminder to P-dubs (paul)
www.themathpage.com/aPreCalc/logarithms.htm
and here is my personalization...
ok this is from arrested development the BEST show ever to air...and probably ever will air

Gob : hey, guy, They tell me you're the actor who plays Marta's brother, Tio
Spanish guy : Como?
Gob : Oh, you're gonna be in a coma alright...

:-D I love that show
ok goodnight everyone

Log Blogg (copyright Mr. french)

7-3: Indentifying Functions from Numerical Patterns

Alright, this lesson is essentially determining the type of function (and it's corresponding pattern) from a set of points (arranged by x and y). It's easiest to break the lesson down by function, so sit back and follow along.

Linear Functions: X and Y values are increased by the addition of a constant amount to each variable. As you can see, we must add 2 for each x value to get to the next, and 3 to each y value to get to the next. Therefore, we refer to this as the "Add-Add Pattern."

X-Y
1-4
3-7
5-10
7-13
9-14


Exponential Functions: A constant value is added to each X value, and a different constant value is multiplied with each Y value. As you can see, 2 is added to every X value and each Y value is multiplied by 9. This is therefore called the "Add-Multiply Pattern."

X-Y
1-15
3-135
5-1215
7-10935

Power Functions: These suck. Just going to throw that out there. Anyways, the majority of X values are multiplied by a constant, just as the majority of Y values are multiplied by a different constant. I say "majority," because certain points simply don't fit the pattern. Go figure. But yeah, as stated earlier, the majority of X values are multiplied by 2 and the majority of y values are multiplied by 8. We ignore the point (9, 3645), because it simply doesn't fit the pattern. This pattern is "Multiply-Multiply"

X-Y
3-135
6-1080
9-3645
12-8640


Quadratic Functions: These are probably the strangest of the bunch. The X values are evenly spaced and added by a constant amount (in this case, 2) and the differences of the Y values are also evenly spaced. Note the "differences" there. Each specific Y value is added by a different amount each time, but they all share a common factor (in this case, 24). So therefore, this is called the "Constant Second-Differences Pattern."

X-Y
1-15
3-5
5-19
7-57
9-119



Example Problem: Determine 1) the type of function and 2) specific pattern from the following information....

X-Y
5-250
10-750
15-2250
20-6750

Solution: Alright, here's what you do. Notice that the X variable is added by a constant value, 5. Now, check out the Y column. These values are multiplied by a constant value, 3. This table therefore reflects an exponential function and the Add-Multiply Pattern.


Okay, we're almost done here. If you want more help, check out this handy website: HERE!

Joe's Personalization = HEREEEEEE

leo, you're next. don't suck

Monday's Quiz Topics and New Posting Order

Here’s a list of topics for Monday’s quiz:

Precalculus Quiz 7.1-3 Topics
Identify types of functions from graphs
Describe characteristics of graphs – increasing/decreasing, concavity
Use patterns to determine the effect on y when x is changed by a certain factor, given a specific relationship between x and y (general and specific cases)
Numerical patterns for linear, quadratic, exponential and power functions
Identify a function from a data set
Determine additional points for a data set
Given a data set and type of function, determine an equation using algebraic techniques.
Predict a data point once equation is determined. Identify if the data point is extrapolated or interpolated.
Demonstrate the validity of a pattern given points in a data set.

That’s it! I’ll be around after school on Friday and in early on Monday morning. If you have questions over the weekend, send me an email and I’ll try to respond Sunday evening.


As promised, here's the new randomly generated new posting order:


Leo
Sam
Paul
Edward
Katie
Nick
Amira
Madison
Christina
Gina
Kaori
Michelle
Joey


In order to attain the impossible, one must attempt the absurd.
- Miguel de Cervantes


Poor study strategies:

Section 7.2: Identifying Functions from graphical patterns

7-2: Identifying Functions From Graphical Patterns
This section is basically a review. It covers the four basic functions (linear, quadratic, power, and exponential) and demonstrates how to find an equation when given a set of coordinates and a function. The functions, with their names and equations, are shown below.

Linear Function
general equation: y=ax+b
parent: y=x
transformed: y-y1=a(x-x1)
Quadratic Function
general: y=ax^2+bx+c (a is not equal to 0 and b and c are contants)
parent: y=x^2
transformed: y-k=a(x-h)^2 (k and h are the coordinates of the vertex)
a>0

a<0 style="color: rgb(0, 0, 0);">When a is positive, the parabola is facing up, like a smile :) and when a is negative, the parabola faces down, like a frown :(

b>1

0<b<1b<0
Power Function
general: y=ax^b (a and b are not equal to 0)
parent: y=x^b
transformed: y=a(x-c)^b+d

Eponential Function
general: y=ab^x
parent: y=b^x
transformed: y=ab^(x-c)+d

b>1

0<b<1

The equation for a logarithmic function with base 10 is y=10^x
The equation for a natural logarithmic function with base e is y=e^x

Here is an example problem:
What is the equation of a power function with the coordinates (6, 151.2) and (4, 44.8)?

The FIRST thing to do is to remember the equation for a power function, which is y=ax^b
then plug the coordinates into the equation. 44.8=a(4)^b and 151.2=a(6)^b. With power functions, you divide the equations to find a and b.

151.2/44.8=a(6)^b/a(4)^b

When you divide the equations, the a cancels out and you can then find the b exponent
you get...

3.375=6^b/4^b= (6/4)^b

3.375=1.5^b

To find the b exponent, take the log of both sides.

log(3.375)=log(1.5)b <---- the b is no longer an exponent b=log(3.375)/log(1.5)

b=3

Now plug b into one of the equations from before.

44.8=a(4)^3
a=44.8/64
a=.7

then you plug a and b into the general power function equation.

y=.7x^3 <--- answer!!

** When finding equations for exponential and power functions, you use this method and divide the equations to find the values of a and b. For quadratic functions, you can use matrices. For linear functions, you simply plug the coordinates into the equation.

Extra help!!!!!
www.vhcc.edu/pc1spring99/basic_function_families/introduction_to_function_families.htm#The%20four%20basic%20functions%20and%20Otheir%20families


There's a new schedule, so good luck to whoever goes next!