Missing Blogs

So these are my missing blogs, sorry I forgot to do them :P

COMBINATIONS AND PERMUTATIONS

Combinations and permutations show how many ways something can be done.

Combinations-used to find the number of orders when the order is not important.

---Formula:

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

Permutations-used to find the number of possible oders when the order of the items are important.

---Formula :

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

In both formulas,

n = total

r = items picked

EX 1. How many ways can 5 different books be arranged on a bookshelf?

The books are different, so order matters. You will use the permutation formula.

5P5 = 5!/(5-5)! = 5! / 0!

= 5*4*3*2*1 / 1

= 120

EX 2. How many ways can you pick a governing council of 3 out of 6 people?

The governing council has 3 equal positions, so the order picked doesn't matter.

6C3 = 6! / [(6-3)! 3!] = 6! / (3!)(3!)

= (6*5*4)/(3*2*1)

= 120 / 6

= 20

STANDARD DEVIATION

The bigger the standard deviation, the more the numbers are spread out.

The smaller the number, the closer the numbers are together.

To find variance and standard deviation, there's this formula that's totally not worth putting in a blog. But to find variance, you take the mean (average) of all the numbers squared and subtract the square of the mean of the numbers.

Variance: mean of squares - square of the mean

To find standard deviation, you take the number you got for the variance and take the square root of it.

EX 1:

Find the variance and standard deviation of 3, 5, 6, 7, and 9.

Mean of squares = 9 +25 + 36 + 49 + 81 = 200

200/5 = 40

Square of mean = 3 + 5 + 6 + 7 + 9 = 30

30/5 = 6, squared is 36

40 - 36 = 4

So variance = 4

Standard deviation = square root of 4 = 2

EX 2:

Find the variance and standard deviation of 1, 7, 9, and 15.

Mean of squares = 1 +49 + 81 + 225 = 356

356/4 = 89

Square of mean = 1 + 7 + 9 + 15 = 32

32/4 = 8, squared is 64

89 - 64 = 25

So variance = 25

Standard deviation = square root of 25 = 5

PROBABILITY

Probability is the likliness of an occurance.

Probability runs between the number 1, meaning it will happen, and 0, meaning it will never happen. It it usually represented as a fraction and reduced.

Probability does not mean that if the probablility is 3/10, it will happen exactly 3 out of 10 times. It just mean that that is the chances of it occuring.

Probability is usually expressed as:

picked / total

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

A: How many red clubs are there?

0/52 = 0

B: How many diamonds are there?

13/52 = 1/4

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

26/52 = 1/2

D: How many cards are black or red?

52/52 = 1

Ex. 2: Rolling two die:

There are 36 possible outcomes in rolling two die. 6*6 = 36

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

COUNTING PRINCIPLE

The counting principle can also be called the multiplication principle. You just multiply the two or more numbers together to find out how many ways something can be done.

Ex1 :

There are 10 runners in a race. They can win first, second, or third place. In how many ways can these runners win each award?

For first place, there are 10 possible runners.

For second place, there are now 9 because one person was in first place.

For third place, there are 8 possible runners.

You multiply them together: 10*9*8 = 720

Ex 2:

How many ways can 9 people stand in a line?

There are 9 spots, 9 for the first spot, 8 for the second, 7 for the third, etc.

Multiplying it all together: 9*8*7*6*5*4*3*2*1 = 362, 880

Ex 3:

How many ways can you make signals with 4 flags? (1 flag, 2 flags... etc.)

1 flag = 4 ways

2 flags = 4*3 = 12

3 flags = 4*3*2 = 24

4 flags = 4*3*2*1 = 24

Add them together, 4 + 12 + 24 +24 = 100

VENN DIAGRAMS

This is going to be awkward to explain.

An upside down U is called an intersection, or or.

A U is called a union, or and.

A line over something denotes "not" and is called the compliment.

-A intersection B = A or B

-Aunion B = A and B

-compliment of A = not A, or everything but A

-compliment of B = not B, or everything but B

-(compliment of A) intersection B = A or B, but not A, which would be only be B

-A intersection compliment of B = A or B, but not B, which would be only be A

-The compliment of (A intersection B) = not (A or B), which would be everything that isn't A or B

-The compliment of (A union B) = not A and B, so all only A's, only B's, and everything else