These are the blogs I was supposed to do while Brob was gone
COMBINATIONS AND PERMUTATIONS
Combinations and permutations show the number of ways something can be done.
Combination- the number of orders when the order of the items is not important.
The Combination Formula:
nCr = (n!) / [(n-r)! r!]
Permutations- the number of possible orders when the order of the items is important.
The Permutation Formula:
nPr = (n!) / (n-r)!
n = total
r = number of items picked
!= factorial
Example: How many ways can you pick a president, a vice president, and a treasurer from a 7 person club?
The positions are different, so order matters. You have to use the permutation formula.
7P3 = 7! / (7-3)! = 7! / 4!
=7*6* 5*4*3*2*1 /4*3*2* 1
=210
Example: How many ways can you pick a governing council of 4 out of 7 people?
The governing council has 4 equal positions, so the order picked does not matter and you have to use the combination formula.
7C4 = 7! / [(7-4)! 4!] = 7! / (3!)(4!)
= 7*6*5*4*3*2*1/4*3*2*1*3*2*1
= 5040/144
= 35
STANDARD DEVIATION
The larger the standard deviation, the more the values are spread out.
The smaller the number, the closer the values are together.
To find variance, you take the mean (average) of all the numbers squared and subtract the square of the mean of the numbers.
There is a formula but its hard to write so the easy way for variance is mean of squares - square of the mean
To find standard deviation, you take the square root the variance.
Example: Find the variance and standard deviation of 1, 6, 9, and 16.
Mean of squares = 1 +36 + 81 + 256 = 374
374/4 = 93.5
Square of mean = 1 + 6 + 9 + 16 = 32
32/4 = 8, squared is 64
93.5 - 64 = 29.4
Variance = 29.4
Standard deviation = square root of 29.4 = 5.4222
Example: Find the variance and standard deviation of 3, 4, 7, 7, and 8.
Mean of squares = 9 + 16 + 49 + 49 + 64 = 187
187/5 = 37.4
Square of mean = 3 + 4 + 7 + 7 + 8 = 29
29/5 = 5.8, squared is 33.64
37.4 – 33.64 = 3.76
Variance = 3.76
Standard deviation = square root of 3.76 = 1.9391
PROBABILITY
Probability is how likely it is that something will happen.
Probability runs between 0, meaning it will never happen, and the number 1, meaning it will always happen. It is usually represented as a reduced fraction.
Probability does not mean that if the probability is 3/10, it will happen exactly 3 out of 10 times. It just means that that is the chances of it happening are 3/10. Probability is not a fortune teller.
Probability is usually expressed as:
Picked / total
Ex. 2: Rolling two six sided dice:
There are 36 possible outcomes in rolling two dice. 6*6 = 36
A. What is the probability of rolling a 5 with two dice?
4/36 = 1/9
B. What is the probability of rolling an even number with two dice?
18/36 = 1/2
Example: In a standard deck of 52 cards:
A: What is the probability of pulling a black heart?
0/52 = 0
B: What is the probability of pulling a club or a heart?
26/52 = 1/2
C: What is the probability of pulling a black or red card?
52/52 = 1
D: What is the probability of pulling a spade?
13/52 = 1/4
COUNTING PRINCIPLE
When using the counting principle, you just multiply the two or more numbers together to find out how many ways something can be done.
Example:
How many ways can 12 people stand in a line?
There are 12 spots, 12 for the first spot, 11 for the second, 10 for the third, and on and on and on.
Multiplying it all together: 12*11*10*9*8*7*6*5*4*3*2*1 = 479,001,600
Example:
There are 11 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 11 possible runners.
For second place, there are now 10 because someone has already taken first place.
For third place, there are 9 possible runners.
You multiply them together: 11*10*9= 990
Example:
How many ways can you make signals with 5 flags? (1 flag, 2 flags... etc.)
1 flag = 5 ways
2 flags = 5*4 = 20
3 flags = 5*4*3 = 60
4 flags = 5*4*3*2 = 120
5 flags = 5*4*3*2*1 =120
Add them together, 5 + 20 + 60 +120 + 120 = 325
VENN DIAGRAMS
A Venn Diagram is a diagram of two or more intersecting circles that show groups that a variable can belong to and the variable can belong to more than one group.
A U is called a union, or and.
An upside down U is called an intersection, or or.
A line over something denotes "not" and is called the compliment.
X union Y = X and Y
X intersection Y = X or Y
compliment of X = not X, or everything but X
compliment of Y = not Y, or everything but Y
(compliment of X) intersection Y = X or Y, but not X, which would be only be Y
X intersection compliment of Y = X or Y, but not Y, which would be only be X
The compliment of (X union Y) = not X and Y, so all only X's, only Y's, and everything else
The compliment of (X intersection Y) = not (X or Y), which would be everything that isn't X or Y
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