Linear Equations and Absolute Value

1. You can add the same number to (or subtract the same number from) both sides of an inequality.
2. You can multiply (or divide) both sides of an inequality by the same positive number.
3. You can multiply (or divide) both sides of an inequality by the same negative number if you reverse the inequality sign.

**You flip the sign of the inequality when you multiply or divide by a negative number.

_______________________________________________________

Ex.1: Absolute value of 3x-9 is greater than or equal to 9
3x-9 is greater than or equal to 9 3x-9 is less than or equal to -9
3x greater or equal 18 3x less or equal 0
x greater or equal 6 x less or equal 0

Ex.2:Solve the equation or inequality.
8x+6>30
subtract 6 from both sides
8x>24
divide both sides by 8
x>3

absolute value inequalities

absolute value of x+3=7
x+3=7           x+3= -7
x=10              x= -4

When you have an absolute value, you have to make an equation set to an even number and another equation set to an odd number.  When you have an absolute value, most of the time you end up with two different answers.  For this equation we have the absolute value of x+3=7.  To solve this, you have to make two separate equations which are x+3=7 and x+3= -7.  Then you just solve for x after and get x=10 and x= -4.

absolute value of 3x-9 is greater than or equal to 9
3x-9 is greater than or equal to 9                     3x-9 is less than or equal to -9
3x greater or equal 18                                       3x less or equal 0
x greater or equal 6                                             x less or equal 0

For this one you aren't given an equal sign, but you solve the same way.  For this problem you set each to a postive and negative 9. The only extra step you have to do is for the negative 9 you have to switch the sign and that makes it less than or equal too.  Then you solve the problem, but you cant forget if you divide by a negative number you have to reverse the sign also.  So when we solve this problem, we get x greater or equal to 6 and x lessor equal to 0.

Reveiw matricies

When multiplying matrices (a)(b)x(b)(c) the inner numbers have to match, and the resulting matrix will be a(c)

Example 1:
Find each matrix product, if it is defined.
[-6 [-1 12 = Not Defined
0] 0 -4]
2x1 2x2
The inner numbers do not match, so it is not defined.

Example 2:
Find each matrix product, if it is defined.
[-2 3 [0 3 = [-18 9
4 2] -6 5] -12 32]
2x2 2x2
-2(0)+3(-6)=-18
-2(3)+3(5)=9
4(0)+2(-6)=-12
4(3)+4(5)=32
Exapmle 3:
Find each matrix product, if it is defined.
[8 -10 [-2 = Not Defined
0 3 -9
-6 4] 1]
3x2 3x1
The inner number do not match, so it is not defined.

Linear Inequalities; Absolute Value!

Hellllo again,
As always, I hope everyone had a wonderful week/weekend!
This past week went by wayyy too fast..
Back to school tomorrow, see you all then!

Linear Inequalities: absolute value of 3x-1 <4 2x+3y>6
Polynomial Inequalities: x^3-x>0 y<3x^2-4x+1

1. You can add the same number to (or subtract the same number from) both sides of an inequality.
2. You can multiply (or divide) both sides of an inequality by the same positive number.
3. You can multiply (or divide) both sides of an inequality by the same negative number if you reverse the inequality sign.

**You flip the sign of the inequality when you multiply or divide by a negative number.

Example 1:
Solve the equation or inequality. If there is no solution, say so.
8x+6>30
subtract 6 from both sides
8x>24
divide both sides by 8
x>3

Example 2:
Solve the equation or inequality. If there is no solution, say so.
8-11x/4<13
8/4 simplifies to 2
2-11/4x<13
subtract 2 from both sides
-11/4x<11
multiply both sides by 4 to cancel the 4 in the fraction out
-11x<44
divide both sides by -11
since you divide by a negative, you flip the sign
x>-4

Combinations and Permutations review

Combinations and permutations are sort of like probability. It is meant to tell how many ways something can be done. There are many different types of Combinations possible to go with many different scenerios. There are also short cuts in finding the combinations. Here are some of the short cuts:

*Combinations-used to find the number of orders when the order is not important. For instance you are arranging 6 items and if three of your items are the same exact thing then you would not count that as three different items when finding the order that you could arrange because they are exactly the same and it would not make any difference.
Combinations formula: x! C y!/x-y! y!

*Permutations-used to find the number of possible oders when the order of the items are important. For instance say you have 5 items that you are rearranging and all five items are different you would have to count each item because it is different and can not be reppeated in the arrangement.
Permutation formula : x! P y!/ x-y!

EX 1. You have 6 kids lining up with 6 different color shirts. Find out how many orders are possible.
Since order is important because all kids have different colors you use permutation.
6! P 1!= 720


EX 2. You have 5 kids lining up to get lunch 3 have the same color shirt. Find how many orders of colors are possible .
Since the order isnt important for 3 of the kids you must use combination.
5! C 3! / 2! 3!= 60/6= 10

Conditional Probability

Formula:

Multiplication Rule: When two events, A and B, are dependent, the probability of both occurring is: P(A and B)= P(A)xP(B|A)

Example- A math teacher gave her class two tests. 25% of the class passed both tests and 42% of the class passed the first test. What percent of those who passed the first test also passed the second test?

		Start with Multiplication Rule 2.
		Divide both sides of equation by P(A).
		Cancel P(A)s on right-hand side of equation.
		Commute the equation.
		We have derived the formula for conditional probability.

Determinants.

So I’m blogging today because I have a volleyball tournament in Golf Shores this weekend and I don’t know when I will be back on Sunday. And for lack of anything to blog about I will blog about Determinants. Missed you Brob, Hope you had a good trip, I heard Jacob did great!

-Notation- similar to absolute value

-If the determinant =0 then it is called a singular matrix because it doesn't have an inverse.

-For a 2X2 matrix you use ad-bc

-For a 3X3 matrix - recopy the first 2 columns, then multiply the diagonals add the right, subtract the left

-For a 4X4 matrix it may be easier to just use your calculator.

Ex.1:

find the determinant of:

2 6

1 9

2(9)-6(1)=12

Ex.2:

find the determinant of:

1 9 6 4

2 5 4 3=-123

7 0 8 6

5 2 9 8

Ex.4:

find the determinant of:

12 3=174

2 15

Ex.5:

find the determinant of:

0 1 2 3

2 0 1 4=25

6 2 3 1

1 0 0 3

--Danielle

Conditional probability

Very upset coach Matt is leaving us but b. rob is coming back. I am very excited(:

" ' " means it does not have

" and" means multiply the two things

Example 1:


a) P(X') = .6
b) P(Y'/X)= .4 x .7 = .28/.4 = .7
c) P(Y'/X')= .5 x .6 = .3/.6 = .5
d) P(X and Y)= .4 x .3 = .12
e) P(X' and Y)= .6 x .5 = .3
f) P(Y)= .12 + .3 = .42
g) P(X/Y)= .12/.42 = .286
h) P(X'/Y)= .3/.42 = .714

Working with Conditional Probability!

Hello everyoneee,
I hope you all have a fun and safe Mardi Gras!
Coach Matt is leaving us now.. those how ever many weeks he was with us went by veryyy fast.. but before he left, one of the many things he taught us was Conditional Probability!
P.S. He was an awesome teacher!
By the way, I hope yall are having fun BRob! I have heard that Jacob is doing great, congrats.
See you whenever you get back!

There weren't very many notes for this section, so this is all I have:

' means does not
"and" means multiply

Example 1:
Find each probability.
X=0.4
Y=0.3
Y'=0.7
X'=0.6
Y=0.5
Y'=0.5
P(X')=.6
P(Y'/X)=.4x.7=.28/4=.7
PY'/X')=.5x.6=.3/.6=.5
P(XandY)=.4x.3=.12
P(X'andY')=.6x.5=.3
P(Y)=.42
P(X/Y)=.4x.3/.42=.286
P(X'/Y)=.6x.5/.42=.714

Example 2:
Find each probability.
R=0.7
J=0.3
K=0.2
L=0.5
S=0.3
J=0.5
K=0.4
L=0.1
P(S)=.3
P(L/R)=.5X.7/.7=.5
P(L/S)=.1X.3/.3=.1
P(JandR)=.3X.7=.21
P(JandS)=.5X.3=.15
P(J)=.21+.15=.36
P(R/J)=.21/.36=.583
P(S/J)=.15/.36=.417

P.S.= we are working with tree diagrams, but I couldn't draw them, so the bolded letters and numbers are what should be on the tree diagrams.

Conditional Probability!

Probability as you is the odds of something happening wheather it favors the event or not. A favorable even is when the number of wanted outcome is greater than 50. Probability is commonly written as a fraction or sometimes even a percentage. In the case of doing mulitile probabilities it is easier just to write things as a decimal, so that is the way its done. To do mulitiple probabilities you must draw a tree.

*Probability tree- A tree drawn to show and or help find probabilities. Can be drawn with decimals, percentages or fractions. It has as many branches for as many kinds of probability you need to find. It has as many sets of branches for as many times that you need to find this probability.

*To make a tree for multiple probabilites you must make sure that the branches in each set add up to 10. And each number of sets under that doubles in count of sets. Still each set must equal 10.

________________________________________________________

Example 1:


a) P(X') = .6
b) P(Y'/X)= .4 x .7 = .28/.4 = .7
c) P(Y'/X')= .5 x .6 = .3/.6 = .5
d) P(X and Y)= .4 x .3 = .12
e) P(X' and Y)= .6 x .5 = .3
f) P(Y)= .12 + .3 = .42
g) P(X/Y)= .12/.42 = .286
h) P(X'/Y)= .3/.42 = .714

equations

8-11x/4 less than or equal to13
x greater than or equal to -4

To solve this problem, we need to solve for x.  The first step to do this, you have to multiply everything in the equation by 4.  You have to get rid of the fraction to start.  Then when you multiply everything by 4, the 4's cancel on one side and the other you just multiply and get 8-11x less than or equal to 52.  Now you have to subtract the 8 on both sides.  Once you do this, you get -11x less than or equal to 44.  The next step to solving this problem, you have to divide -11 on both sides also.  When you divide a number on both sides, you have to flip the sign that is in the middle of the problem.  When you do this, you get x greater than or equal to -4.  This is your answer to the problem.

Variance

Variance

S: Sample
Sigma: Population
E: Sum
X^i: Numbers
N: How many numbers you have
X Bar: Average
S= Sample Standard Deviation

The smaller the number the closer the numbers are together.
The bigger the number the more the numbers are spread out.

Here are the amounts of gold coins the 5 pirates have:

4, 2, 5, 8, 6.

Now, let's calculate the standard deviation:

1. Calculate the mean:

2. Calculate  for each value in the sample:

3. Calculate :

4. Calculate the standard deviation:

The standard deviation for the amounts of gold coins the pirates have is 2.24 gold coins.

Multiple probabilities

Probability as you is the odds of something happening wheather it favors the event or not. A favorable even is when the number of wanted outcome is greater than 50. Probability is commonly written as a fraction or sometimes even a percentage. In the case of doing mulitile probabilities it is easier just to write things as a decimal, so that is the way its done. To do mulitiple probabilities you must draw a tree.

*Probability tree- A tree drawn to show and or help find probabilities. Can be drawn with decimals, percentages or fractions. It has as many branches for as many kinds of probability you need to find. It has as many sets of branches for as many times that you need to find this probability.

*To make a tree for multiple probabilites you must make sure that the branches in each set add up to 10. And each number of sets under that doubles in count of sets. Still each set must equall 10.

Ex1. tree .7 ^ .3 First set. Has two numbers
.7^.3 .7^.3 Second row has two sets with two in each
Suppose that each .7 is the probability for a sunny day. .3 is the probability for a cloudy day.
A. Find the probability of a sunny day the first day. .7
B. Find the probability of both days sunny. .7*.7 =. 49
C. Find the probability of two days with same type of weather. .7*.7 = .49
.3*.3= .9
.9+.49 = .58

Conditional Probibility

So Coach Matt is leaving us and BRob is coming back. Missed you BRob, hope you had a good time! This week we learning conditional probibility:


Example 1:


a) P(X') = .6
b) P(Y'/X)= .4 x .7 = .28/.4 = .7
c) P(Y'/X')= .5 x .6 = .3/.6 = .5
d) P(X and Y)= .4 x .3 = .12
e) P(X' and Y)= .6 x .5 = .3
f) P(Y)= .12 + .3 = .42
g) P(X/Y)= .12/.42 = .286
h) P(X'/Y)= .3/.42 = .714

--Daniellle :)

Variance

This week we learned about variance:

s=sample
sigma=population
the strange E=sum
x^i=numbers
n=how many numbers you have
x bar=average(mean)
s=sample standard deviation

The bigger the deviation, the more the numbers are spread out.
The smaller the number, the closer the numbers are together.

s=square root of E(xi^2)/n-x bar^2
**note that everything is under the square root

Example 1:
Compute the mean, variance, and standard deviation.
The number of grammatical errors on five daily French quizzes:
10, 8, 7, 5, 5
xi xi^2
10 100
8 64
7 49
5 25
5 25
total=263
n=5
x bar=7
xi^2=263
s=square root of 263/5-7^2
=square root of 263/5-49
=square root of 3.6
=1.90
variance=3.6
standard deviation=1.91
mean=7

--Daniellee

Variance

Variance = s = (sum of xi^2)/n - x barred^2.
Standard Deviance is kinda the same.

xi = numbers

n = how many numbers

x barred = mean/average

The variance is the mean of the squares of the deviations from x bar.

Standard deviation- the amount the largest and smallest numbers are away from the mean which is the average.

You find this by adding up all numbers then dividing by the numbers given.

Statistic- a number that describes some characteristic of a set of data.

To find standard deviaton you must use the following formula: s = square root of sum Xi^2 / n - X^2.

N= number of numbers

Xi= given numbers

X=mean of numbers

Example 1:

find standard deviation of 3 4 5 6 7

Xi Xi^2

3 9

4 16

5 25

6 36

7 49

Now add all squared answers and get 135. Now plug into formula.

To find the mean 3+4+5+6+7/5 = 5

sqaure root of 135/5 - 5^2 = square root of 2 = 1.41

Variability!

The variance, written as s^2 or sigma^2, is the mean of the squares of the deviations from x bar.
The standard deviation, denoted s or sigma, is the positive square root of the variance.

s=sample
sigma=population
the strange E=sum
x^i=numbers
n=how many numbers you have
x bar=average(mean)
s=sample standard deviation

The bigger the deviation, the more the numbers are spread out.
The smaller the number, the closer the numbers are together.

s=square root of E(xi^2)/n-x bar^2
**note that everything is under the square root

_____________________________________________________

Ex.1: Find the sample standard deviance and variance for: 3 5 6 7 9

xi xi^2

3 9

5 25

6 36

7 49

9 81

add up all the xi^2 numbers and you get 200

now find the mean by adding 3+5+6+7+9 =30/5 =6

sqaure root of 200/5 - 6^2 = square root of 4 = 2

sample standard deviance=2

variance=4

Variability!

Hope you all had a fantastic weekend!

Well, this week in class we learned about Variability.

The variance, denoted s^2 or sigma^2, is the mean of the squares of the deviations from x bar.
The standard deviation, denoted s or sigma, is the positive square root of the variance.

s=sample
sigma=population
this E looking thing=sum
xi=numbers
n=how many numbers you have
x bar=average(mean)
s=sample standard deviation

The bigger the deviation, the more the numbers are spread out.
The smaller the number, the closer the numbers are together.

s=square root of E(xi^2)/n-x bar^2
**note that everything is under the square root

Example 1:
Compute the mean, variance, and standard deviation.
The number of grammatical errors on five daily French quizzes:
10, 8, 7, 5, 5
xi xi^2
10 100
8 64
7 49
5 25
5 25
total=263
n=5
x bar=7
xi^2=263
s=square root of 263/5-7^2
=square root of 263/5-49
=square root of 3.6
=1.90
variance=3.6
standard deviation=1.91
mean=7

Probability problems solved with combinations

The experiment of choosing 5 cards from 52 and 52C5 equally likely outcomes. Of those, there are 13C5 combinations that contain 5(of the 13) hearts.

P(all hearts) = 13C5/52C5 = 1,287/2,598,960 = 0.000495

Five cards are drawn at random from a standard deck. Find the probability that exactly 2 are hearts.

P(2 hearts and 3 non-hearts) = 13C2 • 39C3/52C5 = 75 • 9139/2,598,960 = 0.274

Three marbles are picked at random from a bag containing 4 red marbles and 5 white marbles. Match each event with its probability.

a. All three marbles are red 4C3 • 5C0/9C3

b. Exactly 2 marbles are red 4C2 • 5C1/9C3

c. Exactly 1 marble is red 4C1 • 5C2/9C3

d. No marble is red 4C0 • 5C3/9C3

sin, cos, and tangent

sin theta/cos theta*tan theta=sin/cos*sin/cos=sin/sin=1

In this problem, we are trying to simplify.  To simplify the problem, we have to find a way to reduce what is given to you throught using formulas and what not.  To even be able to solve this problem, you have to know that tangent is sin/cos.  If you don't know that, you will have a hard time trying to solve the problem.  Other formulas are sec which is 1/cos, csc which is 1/sin, and cot which is cos/sin.  For this problem the only formula you needed to know was tangent.  Now you have to find a way to cancel out things to reduce the example.  So we do sin over cos times sin over cos, which is tangent.  Then we can cancel out the two cos' at the bottom of the faction and now we have sin over sin.  Sin over sin also cancels out and now you are left with 1.  1 is your answer.

Variability

Variability is finding the standard deviation of a set of data and to convert data to standard values.
Standard deviation- the amount the largest and smallest numbers are away from the mean
Mean- average of a given set of numbers. You find this by adding up all numbers then dividing by the numbers of numbers given.
Statistic- a number that describes some characteristic of a set of data.
Median-the middle number in a set of numbers. If given two middle numbers you must find the mean of these to find the median.
Mode- the number or numbers that they are most of in any given set of numbers.

To find standard deviaton you must use the following formula: s = square root of sum Xi^2 / n - X^2.
N= number of numbers
Xi= given numbers
X=mean of numbers

EX1 find standard deviation of 3 5 6 7 9

Xi Xi^2
3 9
5 25
6 36
7 49
9 81
Now add all squared answers and get 200. Now plug into formula.
Get mean 3+5+6+7+9/5 = 6

sqaure root of 200/5 - 6^2 = square root of 4 = 2

Probability

Probability means the odds of something happening or not happening and is also a part of statistics. Probability is always between one and zero. Probability is in a fraction and should be reduced if could do so. Example 1: What is the probability of picking a black heart? 0/52 which reduces to 0. Example 2: What is the probability of picking a red card? 26/52 which reduces to 1/2

This week in class we learned about probability. Probability means the odds of something
happening or not happening and is also a part of statistics. Probability is always between one and zero. Probability is in a fraction and should be reduced if could do so. This
week we mainly worked of probability of a deck of cards and two die being rolled.

Example 1:

What is the probability of picking a black diamond?

Since there is no black diamonds the answer would be 0/52 which reduces to 0.

Example 2:

What is the probability of rolling and even number if you roll
two die?

Since there is to die the possibility is out of 36. There are 6 even numbers so 6/36 reduces to
1/6.

Example 3:

What is the probability of picking a black card?

There are 26 black cards out of 52. So 26/52 which reduces to 1/2.

Probability!

This week we learned about Probability.

Probability is a branch of math that deals with uncertainty. It is the likliness of an occurance. Probability is used for many things like gambiling and meterology.
-Probability can be shown in terms of percentages.
-Probability runs between the number 1 and 0. It it usually represented as a fraction and reduced.
-You can determine the probability either empirically or theoretically. In math we mostly use empirical beacause they are based on exact observations.
-An experiment can be done to determine your own statistics to find the probability of something.

________________________________________________________

Ex.1: In a standard deck of 52 cards:

A: How many diamonds are there?

13/52 or 1/4

B: How many red clubs are there?

0/52 or 0

C: How many cards are black or red?

52/52 or 1

D: What is the probability of pulling a diamond or a heart?

26/52 or 1/2

Probability

Probability is measured from 0 to 1; 0 means no event and 1 means certain event.

Probability of an event: P(event A) = # of favorable outcomes/total # of outcomes

Either of two events: P(A or B) = P(A)+P(B)-P(A and B)

Either of two mutually exclusive events: P(A or B) = P(A)+P(B)

Probabilities!

Helllllooo,
I forgot to do this earlier, so now im stuck doing it during the Super Bowl.. oh wellll.. :D

0=event will not take place
1=certain event

P(event A)= favorable outcomes/total number of outcomes

Probability ranges from 0%-100%

**Prime numbers do not include 0 or 1.

P(A or B)= P(A)+P(B)-P(A and B)
(Probability of either of two events)

P(A or B)= P(A)+P(B)
(Probability of either of two mutuallly exclusive events)

Example 1:
Suppose a card is drawn from a well-shuffled standard deck of 52 cards. Find the probability of drawing each of the following.
a) a black card
=26/52
=1/2
b) spade
=1/4
c) not a spade
=3/4

Example 2:
Suppose a card is drawn from a well-shuffled standard deck of 52 cards. Find the probability of drawing each of the following.
a) red diamond
=13/52
=1/4
b) black diamond
=0/52
=0
c) not a black diamond
=52/52
=1

Example 3:
Suppose a card is drawn from a well-shuffled standard deck of 52 cards. Find the probability of drawing each of the following.
Mr. and Mrs. Smith each bought 10 raffle tickets. Each of their three children bought 4 tickets. If 4280 tickets were sold in all, what is the probability that the grand prize winner is:
a) Mr. or Mrs. Smith
20/4280
=1/214
b) one the of 5 Smith's
=32/4280
=4/535
c) none of the Smith's
=4248/4280
=531/535

Permutation and Combination! KAYLA

permutation=order matters= nPr=n!/(n-r)!
combination=order doesn't matter= nCr=n!/(n-r)!r!
!=factorial which means like 6!=6*5*4*3*2*1

Example 1: a two person governing committee has three people trying out. how many ways can they be picked?
3C2=3*2*1/2*1*1=3

Example 2: three people are trying out for president and vice president. how many ways can they be picked?
3P2=3*2*1/1=6

Probability

Probablity

Probability-A branch of math that deals with uncertainty. It is the likliness of an occurance. Probability is used for many things like gambiling and meterology.
-Probability can be shown in terms of percentages.
-Probability runs between the number 1 and 0. It it usually represented as a fraction and reduced.
-You can determine the probability either empirically or theoretically. In math we mostly use empirical beacause they are based on exact observations.
-An experiment can be done to determine your own statistics to find the probability of something.

Formulas:
*Probability of either of two events: P(A+B) = P(B) + P(A) - P(A and B)
Similar to Compliment formula

*Probability of eith of two mutually exclusive events (means that they happen at the same time) : P(A or B) = P(A) + P(B)

Ex.1If there are 6 apples, 3 oranges, and 1 plum in a basket, what is the probability of choosing an apple without looking in the basket?

Solution:

P(choosing an apple)= 6/10 = 3/5 = 0.6 = 60%

The numerator is 6 because there are six apples in the basket, therefore six favorable outcomes. The denominator 10 because there are 10 fruits in the basket (the total number of outcomes).

Probablity

Probability-A branch of math that deals with uncertainty. It is the likliness of an occurance. Probability is used for many things like gambiling and meterology.
-Probability can be shown in terms of percentages.
-Probability runs between the number 1 and 0. It it usually represented as a fraction and reduced.
-You can determine the probability either empirically or theoretically. In math we mostly use empirical beacause they are based on exact observations.
-An experiment can be done to determine your own statistics to find the probability of something.

Formulas:
*Probability of either of two events: P(A+B) = P(B) + P(A) - P(A and B)
Similar to Compliment formula

*Probability of eith of two mutually exclusive events (means that they happen at the same time) : P(A or B) = P(A) + P(B)

EX1. What is the probability of flipping a quater and getting tails?
1/2.
You get 2 as the bottom number because thats all 2 different possible outcomes. You get one as top number because thats the one outcome you were asked to find.

EX2. What is the probability of rolling a die and getting 3 or 1?
2/6 reduces to 1/3
You have 6 as bottom number because thats all the different possibilities and 2 as you top number because your were asked to find two different outcomes.

Permutations and Combinations

We use combination and permutation to find out how any possible outcomes there are in a situation. 

You use combination when the order is unimportant. 

You use permutation when the order is important. 

Permutation formula: nPr= n!/(n-r)!

Combination formula: nCr= n!/(n-r)!r!

-Permutation Examples

Example 1: Find 7!

Answer: 5040

Example 2: In how many ways can 8 CD's be arranged on a shelf?

Answer: 40320

Example 3: If a softball league has 10 teams, how many different end of the season rankings are possible if there are no ties?

Answer: 3,628,800

Example 4:  In how many ways can a sorority of 20 members select a president, vice president, and treasury, assuming that the same person cannot hold more than one office?

Answer: 6840

Example 5:  How many different arrangements can be made using two of the letters of the word TEXAS if no letter is to be used more than once?

Answer: 20

-Combination Examples

Example 1: In a conference of 9 schools, how many intraconference football games are played during the season if the teams all play each other exactly once?

Answer: 36

Example 2: You are going to draw 4 cards from a standard deck of 52 cards.  How many different 4 card hands are possible? 

Answer: 270,725

Probability

The formula for probability is P(event A)= favorable outcomes/total #of outcomes.  0 means the event will not take place.  1 means certain event.  When you are trying to find the probability of something, you are trying to find its outcome.  For example, P(3 or even numbers) Given numbers 1-6 that you can use, try to solve this problem.  The probability of pulling a 3 out of the numbers 1-6 is 1/6 because there are 6 numbers and only one 3.  The probability of pulling an even number out of 6 numbers is 3/6 because there are still 6 numbers but now there are 3 even numbers out of the selection of numbers we have.  Then for this problem we would add the two answers we got together and you get 4/6, which simplifies to 2/3.

P(3 or even #'s) 1/6 +3/6=4/6=2/3

Probability of Dice and Cards

This week we learned about the probability of rolling dice and of drawing a card from a standard deck of 52 cards.

P(event A) = favorable outcomes/total number of outcomes

Ex 1. A. Find the probability of pulling out a face card out of a standard deck of cards.

12/52 = 3/13

B. Find the probability of pulling out a red card out of a standard deck of cards.

26/42 = 1/2

C. Find the probability of pulling out a spade out of a standard deck of cards.

13/52 = 1/4

D. Find the probability of pulling out a red face card out of a standard deck of cards.

6/52 = 3/26

Ex 2. A. What is the probability of rolling an even number with two dice?

18/36 = 1/2

B. What is the probability of rolling a 5 with two dice?

4/36 = 1/9

Chapter 16 Probability

- 0 to 1 ration: 0 events that will not take place, 1 a certain event
- P(event A) = favorable outcomes/total number of outcomes
- Either of two events: P(A or B) = P(A)+P(B)-P(A and B)
- Either of two mutually exclusive events: P(A or B) = P(A)+P(B)
* Little note: Prime numbers are divisable by one and itself

Example 1: Using a 52 card deck

a) P(black king) = 2/26 which reduces to 1/26

b) P(red face card) = 6/52 which reduces to 3/26

c) P(black or red card) = 52/52 which is 1

d) P(black diamond) = 0/52 which is 0

e) P(spade) = 13/52 reduces to 1/4

f) P(ace) = 4/52 reduces to 1/13


Example 2: Two die

a) P(sum of 2) = 1/36

b) P(the first die will be even number) = 18?36 reduces to 1/2

c) P(sum of -4) = 0/36 which is 0

d) P(factor of 40) = 4/6 reduces to 2/3

---Daniellee

Permutation and Combination

Permutation and Combination are used to find how many possible outcomes.

Permutation is used when order is important

Combination is used when order is unimportant

the formulas are:

nPr = n!/(n-r)!

nCr = n!/(n-r)!r!

! is the factorial symbol which means multiplied by every number less than that number all the way to one. so like 5! would be 5*4*3*2*1

0! = 1

Examples:

1. You have a pile of 5 different looking coins. How many ways can you take 2 coins from the pile?

since the coins are different, order is important and we use permutation

5P2= 5!/(5-2)!

5!/3! =20

2. Twelve people apply for three jobs, if the jobs are all the same how many ways can three people been picked?

since the jobs are all the same, order is unimportant and we use combination

12C3= 12!/(12-3)!3!

12!/9!3!= 1320/3!

220

thats how it is done.