Confidential2 Warm Up 1.Tossing a quarter and a nickel HT, HT, TH, TT; 4 2. Choosing a letter from...
25
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,
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.
Slide 3
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 =
Player 2 = Warm Up
Slide 4
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
Slide 5
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.
Slide 6
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.
Slide 7
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?
Slide 8
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.
Slide 9
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. Lets get Started
Slide 10
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
Slide 11
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
Slide 12
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 22222 = 32
arrangements of heads and tails.
Slide 13
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
Slide 14
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.
Slide 15
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
Slide 16
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
Slide 17
Confidential17 Refreshment Time
Slide 18
Confidential18 Lets play a game
http://www.miniclip.com/games/big-jump-challenge/en/
Slide 19
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.
Slide 20
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
Slide 21
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.
Slide 22
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 = 314457495
There are 314,457,495 different draws.
Slide 23
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.
Slide 24
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)!)
Slide 25
Confidential25 You did great in your lesson today ! Practice
and keep up the good work