Confidential2 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.

Confidential3 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

Confidential4 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

Confidential5 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.

Confidential6 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.

Confidential7 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?

Confidential8 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.

Confidential9 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

Confidential10 Outcomes of an Event Definition: Possible outcomes of an event are the results which may occur from any event. Example: The following are possible outcomes of events : A coin toss has two possible outcomes. The outcomes are "heads" and "tails". Rolling two regular dice, one of them red and one of them blue, has 36 possible outcomes. Note: Probability of an event = number of favorable ways/ total number of ways

Confidential11 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. START H T H T H T

Confidential12 Counting Principle The 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 = 32 arrangements of heads and tails.

Confidential13 Permutation A permutation is an arrangement, a list of all possible permutations of things is a list of all possible arrangements of the things. Permutations are about Ordering. It says the number of permutations of a set of n objects taken r at a time is given by the following formula: nP r = (n!) /(n - r)! Example: A list of all permutations of the letter ABC is ABC, ACB, BAC, BCA, CAB, CBA

Confidential14 Combination Combination 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 n objects taken r’ at a time is given by nC r = (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.

Confidential15 Your turn 1. _______ 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

Confidential16 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. nP r = (n!) /(n - r)!. 10) Write the formula to calculate combination. nC r = (n!) /(r! (n -r)!). Your turn

Confidential17 Refreshment Time

Confidential18 Lets play a game

Confidential19 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 tree diagram to find the number of possible outcomes for the experiment. Solution:- The possible outcomes are RG, RB, GR, GB, BG and BR. So, the number of possible outcomes is 6.

Confidential20 2) The ice cream shop offers 31 flavors. You order a double-scoop cone. In how many different ways can the clerk put the ice cream on the cone if you wanted two different flavors? Solution:- There are 31 flavors available for the first scoop.’ There are then 30 flavors available for the second scoop. The possibilities are = 31 * 30 = 930

Confidential21 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.

Confidential22 n C r = n! / r! (n - r)! Here n = 47 and r = 8 n C r = 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 = There are 314,457,495 different draws.

Confidential23 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.

Confidential24 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)!)

Confidential25 You did great in your lesson today ! Practice and keep up the good work