9.7 Blog Post

9.7: Probability, Dude

Merriam-Webster (rough riders don’t use dictionary.com) defines probability as the ratio
of the number of outcomes in an exhaustive set of equally likely outcomes that produce
a given event to the total number of possible outcomes or the chance that a given event
will occur.

Definin’ Some Termz:

Experiment-activity under consideration
Outcome-Each possible observation
Sample space/Outcome set-set of possible outcomes
Event-subset of a sample space

No Mo Termz

How ‘Bout We Do Some Scenarios???

Lets suppose you’re holding a coin. The type of coin isn’t vital, but lets call it a quarter,
cause quarters are big and yeah. So you’re ready to flip that coin and you ask yourself,
what are the chances that this quarter will land with Mr. Washington looking up at me? If
you are seriously thinking about the answer to this question, you probably need to re-
evaluate your life, but this is just an example so stop nit picking my flawless and
awesome blog post. The point is that the CHANCE you’re trying to calculate is the
PROBABILITY that the quarter will land heads up. You do so by creating a ratio of the
number of favorable outcomes (heads), divided by the number of possible outcomes
(heads and tails). This gives you ½. That’s easy, right? YEAH! But, I doubt Thad will put
it on a test because he likes to use his green pen.

Lets look at a trickier situation, shall we? Say you got Drew’s left shoe, and a six sided
die. The shoe can land either laces up or sole up. If you were to flip the shoe and then
role the die, what is the probability that you get laces and a 2? Lets break it down,
dawg. You gotta ½ chance of flippin’ a lace, right? Of course I’m right. Then, you would
multiply that by the ratio of favorable die outcomes (1) over total possible die outcomes
(6). Thus giving you a 1/12 shot of gettin’ laces and a 2. Done. Next.

Just throwing this out there… P(A)= the probability of A

Almost forgot, mutually exclusive means that two sample spaces do not share a
common event (usin’ dem termz).

Bring in the formulas!!!!!

P(A U B)= P(A) + P(B)*
P(A U B)= P(A) + P(B) - P(A “upside down U” B)
P(A “upside down U” B)= P(A) x P(B)**

Also, independent means that the events don’t influence each other, like how flipping
Barton’s shoe doesn’t effect the roll that the die will land on.

Now we get off track…

http://www.youtube.com/watch?v=eVtDSFgeqPY

http://www.youtube.com/watch?v=jsKpazuC0RY

This is too help Barton and Pete learn how to count:

http://www.youtube.com/watch?feature=endscreen&NR=1&v=85M1yxIcHpw

Back on task! YAY!

Complements are cool too. The complement of event A are all of the possible outcomes
in the sample space that are not in A. I’m tired so I’m not going to explain it in more
detail. The book does an okay job.
A’= Complement A
P(A’)= 1- P(A)

*Only if A&B are mutually exclusive
**Only if A&B are independent

My Closing Statement.

By now you have realized that this is the greatest blog post of all time, followed closely
by all of my other blog post, and then Oran’s famous “I’m going to be as creepy as
possible” post of 2010. I’m going to be completely honest, and I think I speak for
everyone when I say, I am pretty damn impressive.

Pete *****

Barton *****

Seaglass can’t do maths

Austin’s cool

Geller’s too smart

Thad’s a beast

Barton *****

And insert Noah’s blog post here:

I was just too tired to write this blog post. I am sorry that I could not supplement
you with more maths knowledge. Maths is forever.

NNNNNNNNNNNNOOOOOOOOOOOOOAAAAAAAAAAAAAHHHHHH!!!!!!!!!!!

K bye.

9.6 Counting Principles

Fundamental Counting Principle- E₁ and E₂ are two different events. E₁ can occur in m₁ different ways and E₂ can occur in m₂ different ways. The number of different ways that the two events can occur together is m₁ x m₂.

The fundamental counting principle is true for any number of events that occur together.

When counting the number of possible outcomes, it is important to know if you are counting combinations or permutations.

When counting permutations, order matters. This means that choosing option 1, 2, and then 3 is different than choosing option 1, 3, and then 2. Lets look at an example…

If you randomly choose 3 digits (between 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9), where each digit can only be used once, how many numbers can you make?

This is asking for the number of outcomes. Because order matters, we have to count the number of permutations.

According to the fundamental counting principle, we have to find the product of the number of possible outcomes for each event (m₁ x m₂ x m₃).

1. The first digit can be any of the numbers listed above so there are 10 different possible outcomes.
2. The second digit can be any number listed above other than the first digit so there are 9 possible outcomes.
3. There are 8 possible outcomes for the third digit.

So to find the total number of permutations when you choose three out of ten options...

Our equation for counting the number of permutations is...

Where 

Now let’s look at how to count combinations. When counting combinations, order does not matter. This means that choosing option 1, 2, and then 3 is the same as choosing 1, 3, and then 2. Here’s an example of counting the number of possible combinations…

When ordering a pizza with toppings, it does not matter what order you choose the toppings in. A cheese, ham, pineapple pizza is the same thing as a pineapple, ham, cheese pizza. Let’s say there are ten toppings to choose from. How many ways can you choose 3?

If order mattered we would find the answer like this...

However, because order doesn't matter, this number is too big because it counts

cheese, ham, pineapple

cheese, pineapple, ham

ham, cheese, pineapple

ham, pineapple, cheese

pineapple, cheese, ham

and

pineapple, ham, cheese

as separate outcomes, when in reality, they are all the same thing.

To get the answer we want we just divide by 3! or 6. 

We divide by 6 because (as we saw above) any combination of three outcomes has 6 different permutations (because 3!=6)

We can find the number of combinations using...

Remember 

Hope this is helpful!!!

Binomials

Lets expand binomials!!!

WOAH!
      Can you imagine what a pain it would be to
      foil all of those?!

But wait, THERE ARE PATTERNS!

First, notice that as the power of x goes down; the power of y goes up.
Each term follows the pattern (coefficient) x ^n-r y ^r
where n is the power or the whole binomial
    and r is the number of the term minus 1 (for the fourth term, r = 3)

.

Here Be-ith Pascal's Triangle.
Behold the magnificence.
Each row corresponds with the power of the binomial  -- (x + y)ⁿ --  where the top is n = 0, second n = 1, third n = 2, etc. Each number in the row corresponds with the coefficient of each of the terms of the expanded binomial.
(Compare the binomials' coefficients to the numbers in the triangle at the right. It will make sense.)
One can find the coefficient by using the number in the triangle from the n^th row and the r^th diagonal.
.
One can find the numbers in the next row by adding together the two numbers directly above it in the previous row.  Unfortunately, Pascal's Triangle, although an incredible feat of wizardry, is not the most effective method. Especially when dealing with the eighty somethin-ith term.

You can figure out what the coefficient of any term is by using
The Binomial Theorem. It's pretty great.

If the terms of the compounded binomial already HAVE coefficients, you put THAT to the powers of n-r and r also and then multiply it by the coefficient you get from compounding it.

That's it, bloggy friends.
Olivia Miller

Chapter 9.4 Mathematical Induction

                                                          Chapter 9.4 Mathematical Induction

What is Mathematical induction???????

Mathematical induction is a method of proof in which a statement is proved for one step in a process, and it is shown that if the statement holds for that step, it holds for the next.

The reasoning behind mathematical induction can be linked to a infinite chain of dominos.  It is impossible to go and push the dominos down one by one.  However, if it is given that by pushing down one domino it will push down that domino next to it then it can be inferred that simply by pushing down one domino you are pushing them all down.

It is very important to note that this process must be a legitimate proof, and is not simply jumping to conclusions.

Mathematical induction is a three step process.

1. Show that the claim is true for n=1

2. Assume that the claim is true for n.

3. Show that it is true for n+1

Example

1 + 3 + 5 + ... + (2n-1) = n²

1. Show it is true for n=1

1 = 1² is True

2. Assume it is true for n=k

1 + 3 + 5 + ... + (2k-1) = k² is True

3. Now, prove it is true for k+1

1 + 3 + 5 + ... + (2k-1) + (2(k+1)-1) = (k+1)² 

 1 + 3 + 5 + ... + (2k-1) = k² (the assumption above), so we can do a replacement for all but the last term:

k² + (2(k+1)-1) = (k+1)²

Now expand all terms:

k² + 2k + 2 - 1 = k² + 2k+1

And simplify:

k² + 2k + 1 = k² + 2k + 1

It's good!

So:

1 + 3 + 5 + ... + (2(k+1)-1) = (k+1)² is True

And that's mathematical induction In a nut shell.

www.youtube.com/watch?v=lOGqZAMHS-M

Matthew Silbergleit

Partial Faction Decomposition (Included in 7.3)

Partial fraction decomposition is essential to success in Calculus next year.  PFD involves breaking down a rational faction with a polynomial denominator into the sums of smaller fractions.

Before "decomposing" a fraction, you first have to take the degrees of the numerator and denominator of the rational function into consideration.

• If the degree of the numerator is greater than that of the denominator (improper fraction), divide the numerator into the denominator by long division.  This will result in a polynomial followed by the remainder of the long division divided by the original denominator.  Then proceed to decompose the latter using the steps that will be explained later.
• If the degree of the numerator and the denominator equal, follow the steps above and use long division to break the function down further.
• If the degree of the numerator is less than that of the denominator (proper fraction), proceed using the steps below.

Here is an example:

   First, factor the denominator. 

Next, make this fraction equal to a sum of two fractions with the factored denominators.

Working with the right side, proceed as you would to find a common denominator to add the fractions together.  Add the like terms and begin to simplify the equation.














As you can see the  term has a coefficient of 1 and therefore  equals 1.  Consequently,  must then equal 5.  Create a system of equations from here and solve.







Your final answer:




Side note- In cases where the denominator has repeated factors such as , be sure to decompose as followed:





Thanks for reading!

Julia Wilkins

9.2: Arithmetic Sequences & Partial Sums

As defined by the book, a sequence is arithmetic if the differences between consecutive terms are the same.

The common difference, or d, is the difference between each number in an arithmetic sequence.

An example of an arithmetic sequence would be something like this:

3, 5, 7, 9, 11, 13, 15, ...

This sequence has a common difference is 2.

There are a few equations that go along with arithmetic sequences and this section.                                                         First is the explicit formula:
                                  

         

Next is the recursive formula:

To find the nth term of an arithmetic sequence, use the formula:

Last is the formula to find the partial sum (the sum of the terms in a sequence):

I think that's about it for arithmetic sequences! And remember, never go in against a Sicilian, when death is on the line!

Jessica

           

9.1 Summation Notation

Summation notation is an easy way to represent the addition of the terms of a finite, or restricted, sequence. It uses the Greek letter Sigma.
 

In summation notation the first term of the sequence being added goes below the Sigma while the last term being added goes above.


Below is an example of summation notation:
 

In this case, the first term being added is the third term, because there is a 3 below the Sigma. The last will be the seventh, due to the 7 above the Sigma. The total will be the sum of the 3rd, 4th, 5th, 6th and 7th terms of the sequence.


Product Notation

While summation notation is the addition of the terms of a sequence, product notation is the product of the terms of a sequence multiplied together.  It uses the Greek letter Pi.



Like summation notation, the first term in of the sequence being multiplied goes below the Pi while the last term goes above.

 

In this case the 5th, 6th and 7th terms will be multiplied together to find the product of this portion of the sequence.



One Dozen Monkeys.
Which is always a let down, but for the sake of tradition must go on the blog.
http://www.youtube.com/watch?v=3nG1ckY7thw



Also you should know sun bears are cool




You already know Levitsky.

Beware the Wheel.

Don't be a book licker.

-Barton

9.1: Sequences and Summation Notation

Let's start with some basic definitions

• A Sequence is an ordered list of numbers
• Sequences can be written as an Explicit Formula or a Recursive Formula
• the Explicit Formula is used to determine any term a_n in a sequence.  Here is an example

Lets compare this to a function

 See that the two are very similar, but in a function, the domain is all real numbers.  In our sequence, the domain is natural numbers. 

•  the Recursive Formula is used to determine the next number in a sequence, or k=1, when given k

Notice that along with the initial equation, one is given one value in the sequence, here a₁, called the Initial Condition. 

Here are some handy patterns to know and be able to recognize

Lastly, Lets go over a short summary of Summation Notation using sigma.  when one wishes to add a certain number of terms in a sequence, it is more convenient to use Summation Notation than to write out each term.  Here is an example:

Where the number below the sigma is the starting term in the sequence, and the number on top is the ending term.  a_i is the explicit formula for the sequence.  

The terms in a sequence can also be multiplied using a similar notation

 

 This has been a general overview and review of Sequences and Summation Notation.

Vote Erin for NHS President! :)