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Warm Up

1. Tossing a quarter and a nickel HT, HT, TH, TT; 4

2. Choosing a letter from D,E, and F, and a number from 1 and 2

D1, D2, E1, E2, F1, F2; 6

3. Choosing a tuna, ham, or egg sandwich and chips, fries, or salad

TC, TF, TS, HC, HF, HS, EC, EF, ES; 9

For each situation, list the total number of outcomes.

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Determine whether the following game for two players is fair.

4. Toss three pennies. No

5. If exactly two pennies match, Player 1 wins. Otherwise, Player 2 wins. Player 1 = ¾ Player 2 = ¼

Warm Up

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Lets recap what we have learned in this

lesson There are two basic types of trees. • Unordered Tree• Ordered Tree

In an unordered tree, a tree is a tree in a purely structural sense

A tree on which an order is imposed — ordered Tree

A node may contain a value or a condition or represents a separate data structure or a tree of its own.

Each node in a tree has zero or more child nodes, which are below it in the tree

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A Sub tree is a portion of a tree data structure that can be viewed as a complete tree in itself

A Forest is an ordered set of ordered trees

Traversal of Trees

• In order• Preorder• Post order

In graph theory, a tree is a connected acyclic graph.

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Preorder And Post order Walk

• A walk in which each parent node is traversed before its children is called a pre-order walk;

• A walk in which the children are traversed before their respective parents are traversed is called a post-order walk.

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Tree diagram

The probability of any outcome in the sample space is the product (multiply) of all possibilities along the path that represents that outcome on the tree diagram. A probability tree diagram shows all the possible events.

Example: A family has three children.  How many outcomes are in the sample space that indicates the sex of the children? 

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There are 8 outcomes in the sample space. The probability of each outcome is 1/2 • 1/2 • 1/2 = 1/8.

Assume that the probability of male

(M) and the probability of female

(F) are each 1/2.

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Sample space

Sample space is the set of all possible outcomes for an experiment. An Event is an experiment.

Example:

1) Find the sample space of rolling a die.

Sample space = { 1, 2, 3, 4, 5, 6 }

2) Find the sample space of Drawing a card from a standard deck.

Sample space = { 52 cards}

3) Rolling a die, tossing a coins are events.

Let’s get Started

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Outcomes of an Event Definition:

Possible outcomes of an event are the results whichmay occur from any event.Example:

The following are possible outcomes of events :• A coin toss has two possible outcomes. The outcomesare "heads" and "tails".

• Rolling two regular dice, one of them red and one ofthem blue, has 36 possible outcomes.

Note: Probability of an event = number of favorable ways/ total number of ways

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Example

If two coins are tossed simultaneously then the possible

outcomes are 4. The possible outcomes are HH, HT, TH,

TT. The tree diagram below shows the possible outcomes.

STARTH

T

H

TH

T

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Counting PrincipleThe Counting Principle is MULTIPLY the number of ways each activity can occur. If event M can occur in m ways and is followed by event N that can occur in n ways, then the event M followed by N can occur in m x n ways.

Example:

A coin is tossed five times.  How many arrangements of heads and tails are possible?

Solution:

By the Counting Principle, the sample space (all possible arrangements) will be 2•2•2•2•2 = 32 arrangements of heads and tails.

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PermutationA permutation is an arrangement, a list of all possible

permutations of things is a list of all possiblearrangements of the things. Permutations are aboutOrdering. It says the number of permutations of a setof n objects taken r at a time is given by the followingformula: nPr = (n!) /(n - r)!  

Example: A list of all permutations of the letter ABC is

ABC, ACB, BAC, BCA, CAB, CBA

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CombinationCombination means selection of things. The word

selection is used, when the order of things has no

importance. The number of combinations of a set of nobjects taken r’ at a time is given by

nCr = (n!) /(r! (n -r)!)  

Example:

4 people are chosen at random from a group of 10

people. How many ways can this be done?

Solution: n= 10 and r = 4 plug in the values in the formula There are 210 different groups of people you can choose.

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Your turn1. _______ diagram shows all the possible events.

Tree diagram

2. Write the possible outcome if a coin is tossed?

{H, T}

3. ________is MULTIPLY the number of ways each activity can occur. Counting principle

4. How many elements are in the sample space of tossing 3 pennies? 8

5. A _______ is the set of all possible outcomes. Sample space

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6) _______ is an experiment. [event, outcome] event

7) ____________ is an arrangement. Permutation

8) Combination means _________ of things. Selection

9) Write the formula to find permutation.nPr = (n!) /(n - r)!. 

10) Write the formula to calculate combination.nCr = (n!) /(r! (n -r)!).  

Your turn

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Refreshment Time

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Lets play a game

http://www.miniclip.com/games/big-jump-challenge/en/

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1) A box has 1 red ball, 1 green ball and 1 blue ball, 2 balls are drawn from the box one after the other,without replacing the first ball drawn. Use the treediagram to find the number of possible outcomes forthe experiment .Solution:-• The possible outcomes are RG, RB, GR, GB, BG and BR. • So, the number of possible outcomes is 6.

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2) The ice cream shop offers 31 flavors.  You order adouble-scoop cone.  In how many different ways can

theclerk put the ice cream on the cone if you wanted twodifferent flavors?

Solution:-There are 31 flavors available for the first scoop.’There are then 30 flavors available for the secondscoop.The possibilities are = 31 * 30 = 930

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3) 8 students names will be drawn at random from a hat containing 14 freshmen names, 15  sophomore names, 8 junior names, and 10 senior names. How many different draws of 8 names are there overall?Solution:-

This would be a combination problem, because a draw would be a group of names without regard to order. 

There are 14 freshmen names, 15  sophomore names, 8 junior names, and 10 senior names for a total of 47 names.

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

Here n = 47 and r = 8 n Cr = 47! / 8! (47-8)!

= 47 * 46*45*44*43*42*41*40*39! 8! 39!

= 47 * 46*45*44*43*42*41*40*39 8 * 7* 6 * 5 * 4 * 3 * 2 * 1

= 314457495

There are 314,457,495 different draws.

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Event : An Event is an experiment.

Outcome: Possible outcomes of an event are

the results which may occur from any

event.

Lets review what we have

learned in our lesson

Counting principle: Counting principles

describe the total number of possibilities or

choices for certain selections.

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Permutation:

A permutation is an arrangement. Permutations are about Ordering.

The formula is nPr = (n!) /(n - r)!

Combination:

Combination means selection of things. Order of things has no importance.

The formula is nCr = (n!) /(r! (n -r)!)

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You did great in your lesson today !

Practice and keep up the good work