Lesson 66- Experimental and Theoretical Probability

Theoretical Probability:
Example 1:

a)What is the theoretical probability of getting a 5 when rolling a die?
Outcomes: 1,2,3,4,5,6 < Total Outcomes (all)
Favorable outcome:  P(5) = 1/6 < Theoretical

b) Simulate rolling a die 300 times and recording the number of times a 5 appears.
For this calculation you have to use the Randbin
Randbin (1, 1/6,300) = 56/300

Example 2:
You can also use a tree diagram to calculate your answer.
What is the probability that in a family of 3 there are exactly 2 girls?
Tree Diagram:

Determine the number of branches that contain 2 girls?
P(2G)= 3/8 = 37.5%
 * find the branches that show only two girls

Probability:
If an experiment has n equally likely outcomes of witch r outcomes are favourable to event A, then the probability of event A is:  
                                               P(a) = r/n

Lesson 63- 65

Practice Test, and Chapter 6 test

Lesson 62- The Binomial Theorem

Expand and simplify each of the following powers of the binomial a + b:
(a + b)^1 = 1a + b^1
(a + b)^2 = 1a^2 + 2ab + 1b^2

Powers of a: decreasing
Powers of b: increasing

Example 1: (2x + 1)^4
= 4C0 (2x)^4(1)^0 + 4C1 (2x)^3 (1)^1 + 4C2 (2x)^2 (1)^2 + 4C3 (2x)^1(1)^3 + 4C4 (2x)^0 (1)^4
=  16x^4 + 32x^3 + 24x^2 + 8x + 1

Binomial Throrem using combinations:
( a + b)^n = nC0 (a)^n (b)^0 

Lesson 61- Pascal's Triangle

General Triangle:

    The Symmetrical Pattern:
    5C1 = 5C4
  Justification: by symmetry   choosing 1 out is choosing 4 in
 Generalization: nCr = nCn-1


The Recursive Pattern:
5C3 = 4C2 + 4C3
Justification:  add above 2 terms
Generalization: nCr = n-1Cr-1 + n-1Cr
  

Lesson 60- Combinations

Combination's  have no order and Permutations do have an order.

Example 1:
If 5 sprinters compete in a race, how many different ways can the medals for first, second and third place be awarded?
5 x 4 x 3 = 5p3 = 60  genine
This is an example of a permutation of 5 objects taken 3 at a time

If 5 sprinters compete in a race and the fastest 3 qualify for the relay team, how many different relay teams can be formed?
YYY NN     Yes: succeed   No: fail          5!/ 3! 2! = 120/ 12  = 10
This is an example of a combination of 5 objects taken 3 at a time

Combinations:
An unordered arrangement if distinct objects is called a combination
The number if combination's of n distinct objects taken r at a time is nCr

Lesson 59 - Permutations involving identical objects

What happens if two of the three letters are the same?
Take the number of total letters and divide it by the number of repeated letters

For example the word OGOPOGO:
There are 7 letters all together, and two letters which have repeats.
the equation for the permutations is 7!/ 4! 2!        4 - o and 2- g
which equals  105

The number of permutations of n objects of which there are a objects alike of one kind, b alike of another kind, c alike of another kind, and so on.          N!
                                                                  A!B!C!    

Example 2:
A true-false test has 7 questions. How many answer keys are possible if 3 answers are T and 4 answers are F? 
TTT FFFF          7!   =      5040    =  35
                       3!  4!            144

Example 3 :
On the following grid, how many different paths can A take to get to B, assuming one can only travel east and south?

Lesson 58 - Permutations Involving different objects

Investigate:
Two letters, A and B, can be written in two different orders, AB and BA, These are permutations or A and B.
a) List all of the permutations of 3 letters A, B , and C.
ABC, ACB, BAC, BCA, CAB, CBA
 3  x 2 x 1 = 6
How many permutations are there?
    6

b) List all of the permutations of 4 letters A, B, C, and D
ABCD, ABDC, ACBD, ACDB, ADBC, ADCB
BCAD, BCDA, BADC, BACD, BDAC, BDCA
  How many permutations are there?
  4 x 3 x 2 x 1= 24

c) Predict the number of permutations of 5 letters A, B, C, D, and E
  5 x 4 x 3 x 2 x 1 = 5!
! : 5 4 3 2 1

Example 1:
When you press the "scramble" button on a cd player it plays a permutation of the songs on the CD. If the cd has 5 songs on it, how many permutations of the songs possible?

5! or 5p5 = 120                5p5: 5 songs ,  p: means how many your picking, you are picking 5


Factorial Notation:
1! = 1
2!= 2 x 1
3!= 3 x 2 x 1
n!= n (n-1)(n-2)(n-3) ......1

An ordered arrangement of distinct objects is called a permutation
The number of permutations of n distinct objects is n!
The number of permutations of n distinct objects taken r at a time is npr

Example 2:
np2= n!/ (n-2)!    
n(n-1)(n-2)!/ (n-2)!
= n(n-1)