9.7 Probability

9.7 Probability

 

Definitions

An experiment is an activity under consideration.
ie: tossing a coin or rolling a die.

Each possible observation in an experiment is an outcome.

The set of possible outcomes of an experiment is called sample space / outcome set.
ie: the sample set when rolling a die is

Any subset of a sample space is called an event.

Multiple Experiments

If the events are independent, the outcome of one does not influence the other.
ie: rolling a die does not effect the outcome of flipping a coin

Events are mutually exclusive if they cannot both occur.
ie: -when rolling a die one time, rolling an even and odd number are mutually exclusive events
     -When picking a marble from a bag with three blue marbles and three red marbles, picking a red or blue marble are mutually exclusive

Probability of an Event

P(E)= The probability of an event.
n(E)= The number of outcomes of event E
n(S)= The number of outcomes in the sample space S.


     

where  




When P(E)=0, the event is an impossible event and cannot occur.
When P(E)=1, the even is a certain event and must occur.

Probability of the Union of Two Events

 -Means the intersection of two sets.
 -AKA "and"

-Means the union of two sets
-AKA "or"

When A and B are mutually exlusive (have no intersects):

     


When A and B aren't mutually exclusive (will work for all cases though):

Example: In this experiment you are rolling a die one time. What is the probability that the number rolled on the die is either even or prime?

The A circle of the venn diagram has all even numbers (2,4,6) and the B circle of the venn diagram has all prime numbers (2,3,5). 2 is common to both A and B. (1) is not a part of either circle because it is neither even nor prime.

                    

So the probability of rolling either an even or prime number is 5/6                                                                
                                 

Probability of Independent Events

If A and B are independent events, the probabilty that they will both occur is:




Example: You have ten socks in a drawer. 4 are blue, 5 are red, and 1 is green.  What is the probability that if you pick out 2 socks at random, what is the probability that the first sock will be blue and the second one green?

The probability of this happening is 2 in 45.

Probability of a Complement

If A is an event and is its complement then,

 

 

All done,
    Hannah

9.6 Counting

9.6: Counting

Counting is a very useful skill that can be applied to solving probability problems. Counting allows you to find out the number of possible ways something can occur.

The simplest counting problems can be solved by listing out all of the possible outcomes of a given situation. For example, you are given 8 pieces of paper numbered 1 to 8 and asked how many combinations of two papers will add up to 12. To solve this problem, you could simple list out all of the different combinations of numbers between 1 and 8 that add up to twelve. By making a table like the one below, you can count that there are 5 different ways of obtaining a sum of 12.

First Number	4	5	6	7	8
Second Number	8	7	6	5	4

However, most problems aren’t this simple. When a problem has hundreds or thousands of different outcomes, writing out all of the possible outcomes becomes an issue.

One way of solving such problems is to use the Fundamental Counting Principle.

Put simply, the fundamental counting principle states that the number of ways that two events can occur is (Number of ways the first event can occur) x (number of ways the second event can occur, after the first event has already occurred), or (m₁) (m₂).

The fundamental counting principle can also be applied to more than just two events. For example, you could have (m₁)(m₂)(m₃)

Sample problem: How many different combinations of letters and numbers are there for license plates if they always follow the pattern of LNNNLL? (L = letter, N=number)

Using the fundamental counting principle, you simply multiply together the amount of ways each of the 6 characters on the license plate can occur. Since there are 26 letters in the alphabet, and 9 digits, you can find the answer by multiplying (26)(9)(9)(9)(26)(26).

A Permutation is an ordering of elements where such that order matters.  For example, in a permutation, ABC is considered a different combination than CBA.

The equation for finding the number of permutations of n elements is simply n!

Sample problem: How many permutations are there for the letters A, B, C, D, E, and F?

To solve this, you simply do 6! because there are 6 elements. To help better understand this, think of it step by step. For the first element, you have 6 choices. Then for the second one, you have 5 choices since on of your elements was used up in the first step. Then for the third you have 4 choices, and so on.

Finding the number of permutations is not always so simple. Sometimes, you have to order subsets of elements, rather than just ordering one element at a time. You can do this by using the equation for _nP_r, which simply mean n elements chosen r at a time. The equation for this is _nP_r =

Sample Problem: In a race of 12 people, how many ways can these people come in first, second, and third?

To solve this, you must first identify that you are choosing from 12 people, 3 at a time (first, second, and third). So, you know that you can replace _nP_r with ₁₂P₃, which you can then plug in. You can either solve this using the equation, or type it into your calculator.

Sometimes, you will need to find the number of distinguishable permutations. You need to do this when you have elements that are the same. The equation for this is

What you are dividing by is the amount of elements that are the same to eliminate identical permutations. This can be easily demonstrated in the next sample problem.

Sample Problem: How many distinguishable ways can the letters of WISCONSIN by written?

All you do to solve this is plug it into the above equation so you get

In this problem, n=9 because WISCONSIN has 9 letters. On the bottom, you have 2! once because there are two I’s in the word, and again because there are two S’s in the word. So, n₁ and n₂ and so on are just the number of times a certain element is repeated. 

A Combination is the ordering of elements when order DOESN’T matter. So, ABC would be considered the same as CBA.

When looking for the number of combinations of n elements, the approach is similar to finding permutations. To begin, you plug information into _nC_r, or n elements chosen r at a time. From there, you simply plug n and r into the equation _nC_r =

If you have a calculator, you can just type in _nC_r to find the number of combinations. This operation can be found in the probability section of the MATH key.

Sample problem: How many different ways can 4 digits be chosen if the order of the digits doesn’t matter?

Since you have a total of 9 digits, which you are choosing 4 at a time, you can come up with the equation for ₉C₄. You simply plug this in to find your answer.

So, counting really isn’t that hard you just have to think and know which equation to use. Plus your calculator can help you a ton. So yeah. That’s counting.

-Corin 

9.5: The Binomial Theorem

Hello folks, let's learn about the Binomial Theorem....

So a binomial is just a polynomial that has two terms, like so:



Where n is any number.  The Binomial Theorem helps you calculate binomial coefficients; it provides a quick way to raise a power of a binomial.

Let's find some coefficients!

When you expand

 , you get this:



Each term has three parts; the binomial coefficient, the x value, and the y value.  We'll bother about the x and y values of each term in a second, but first here's how to find the coefficient of each term, or nCr.

The coefficient of             is:




....nCr can also be written as :

 

Here's an example:
 


**In general, it is true that    

Pascal's Triangle

Pascal's Triangle is a VERY convenient way to remember a pattern of coefficients. 


The first and last number of each row is 1, and every other number in each row is formed by adding the two numbers above the number:

 

Pascal's Triangle shows the coefficients for each term in the expansion of a binomial...

The top row of Pascal's Triangle is called the zero row, and it is the expansion of the binomial


The next row is called the first row; it is the expansion of


Following suit, the next row is called the second row, and it shows the coefficients of each term in the expansion of


In general, the nth row in Pascal's Triangle gives the coefficients of    

X and Y values of each term in the binomial expansion

To find the x values in each term, you start out with , then go  for the second term, and  for the third term.  The expansion always ends with , which is one.

To find the y values in each term, start with , which is one.  The next term's y value is , and the third term is .  Keep adding one to the power when you find the next term, and the final term should be .

It's quite simple, really!!

    
Let's try one:

write the expansion for the expression

:

The binomial coefficients from the fourth row of Pascal's Triangle are 1, 4, 6, 4, 1. 

--For expansions of binomials representing differences, rather than sums, you alternate signs--

Therefore,



Now, simplify:




And there you have it boys and girls! The Binomial Theorem! 

Henry

9.1 Intro to Sequences

I know this is a week late, but better late than never!

This section is on the basics of sequences. But what is a sequence?

Sequence: function whose domain is an ordered set of natural numbers

There are two ways to define sequences.

Explicit form:
Will tell you any term



The first five terms of the sequence are...

n	1	2	3	4	5
an	1	3	5	7	9

Recursive form: 
All terms are defined using previous terms

a₁ = 1

a_k+1 = a_k + 2

The first five terms of the sequence are...

k	1	2	3	4	5
ak	1	3	5	7	9

It's the same sequence as before, but it's written differently. 

Some common sequences we may see are....

n²: 1, 4, 9, 16, 25…

n³: 1, 8, 27, 64, 125…

2ⁿ: 2, 4, 8, 16, 32…

3ⁿ: 3, 9, 27, 81…

n! (factorial): 1, 2, 6, 24, 120…

Fibonacci: 1, 1, 2, 3, 5, 8, 13, 21…

So, it wasn't very long but that's the introduction to sequences!

Carly

9.2 Arithmetic Sequences and Partial Sums

Wahoo! Sequences! Yay! (Please note the sarcasm. Sorry Mr. Wilhelm)

Now, what could an arithmetic sequence be... a fun phrase to say? Or maybe a type of dessert...probably not. It's a sequence whose consecutive terms have a common difference. Or, the more math way of putting that:

 and so on.

But how does one write that?  where d is the common difference, 

and c is 

Example: 

    7,11,15,19

       Hmm...11-7 is...4. So that means 4 is d. To find c would be 7-4 and that's...3. So the formula would be 4n+3. So to find any term, be it the fifth or the five thousandth, you just plug the term in for n. 

The recursive formula is  or . 

Example:

    Find the seventh term of the arithmetic sequence whose first two terms are 2 and 9. 

        Step 1) In order to find the seventh term, we need to find the formula for the nth term. Since the first term is 2, that means that .  That means the formula for the nth term is . 

        Step 2) Since we are trying to find the seventh term, plug 7 in for n. 

        Step 3) Math. If you did it correctly, which is hard sometimes, you should get the seventh term to be 44. If not...recheck your basic math. That's always a rough spot. 

Woah there, were finding sums now? Jeez.

The Sum of a Finite Arithmetic Sequence  . Beware, this formula ONLY WORKS FOR ARITHMETIC SEQUENCES.

Example 1:

    Find the sum of integers 1 to 100. (Seems scary right? Wrong!)

        Step 1) What's n? 100

        Step 2) What's the first term? 1. What's the last term? 100.

        Step 3) Math. Again, if basic algebra serves you well, you should get 5050.

Example 2:

    Find the sum 1+3+5+7+9+11+13+15+17+19.

        Step 1) What's the common difference? 2.

        Step 2) How many terms are in the sequence? 10.

        Step 3) Plug away! You should find the sum to be 100. 

So, we've found the sum of an arithmetic sequence, but what about the partial sum, eh?

Don't worry, it's the same formula as the sum of an arithmetic sequence. The only difference is that this sequence goes on forever. But you do the same steps, find the value of the n. Plug and chug away!

I think that about covers it. If you have any questions, refer to your neighbors, friends, be a booklicker, or even ask all mighty Mr. Wilhelm himself.

Could it be? My last Wilhelm blog post? Nooooooo :(

-Kristy