Probability and Statistics

1
Probability and Statistics

• Teacher Quality Grant

2
(No Transcript)
3
(No Transcript)
4
(No Transcript)
5
(No Transcript)
6
(No Transcript)
7
(No Transcript)
8
What do you know about probability?

• Probability is a number from 0 to 1 that tells
  you how likely something is to happen.
• Probability can have two approaches -
  Experimental probability
• - Theoretical probability

9
Key Words

• Experimental probability
• Theoretical probability
• Law of Large Numbers
• Outcome
• Event
• Random
• Click here to check the words

10
Definition of Probability

• Probability is a measure of how likely it is for
  an event to happen.
• The probability of one or more events is a number
  between 1 and 0
• The notation for and event is P(event)

11
Probability
P0
P1
Event will never occur
Event is equally likely to occur or not occur
Event will always occur
12
Chance / Probability

• When a meteorologist states that the chance of
  rain is 50, the meteorologist is saying that it
  is equally likely to rain or not to rain. If the
  chance of rain rises to 80, it is more likely to
  rain. If the chance drops to 20, then it may
  rain, but it probably will not rain.

13
Probability Activity 1

• The probability of an event happening can be
  expressed as a percent between 0 and 100. The
  probability of an event that is impossible is
  expressed as 0. The probability of an event that
  is certain to happen is expressed as 100.

14
Probability Activity 1

• The table below shows a seven-day weather
  forecast, including the probability of
  precipitation (POP). The event in this case is
  rain, and the probability is a number expressed
  as a percent between 0 and 100.

15
Probability Activity 1

• 1. For which days does the forecast indicate no
  possibility of rain?
• Sunday, Monday, and Saturday
• 2. For which day does the forecast indicate that
  rain is as likely to happen as not?
• Wednesday
• 3. On which day is it more likely to rain,
  Thursday or Friday?
• Friday
• 4. On which day is it less likely to rain,
  Tuesday or Thursday?
• Tuesday

16
Probability Activity 1

• 5. Discuss whether the amount of rain on Tuesday
  (POP 15) could be greater than the amount of
  rain on Thursday (POP 20), assuming that it
  rains on both days.

17
Experimental vs. Theoretical

• Experimental probability
• P(event) number of times event occurs
• total number of trials
• Theoretical probability
• P(E) number of favorable outcomes
  total number of possible outcomes

18
THEORETICAL PROBABILITY
The theoretical probability of an event is often
simply called the probability of the event.
When all outcomes are equally likely,
the theoretical probability that an event A
will occur is
number of outcomes in A
P (A)
outcomes in event A
19
Theoretical probability

• P(head) 1/2
• P(tail) 1/2
• Since there are only two outcomes, you have 50/50
  chance to get a head or a tail.

20
Theoretical probability
1. What is the probability that the spinner will
stop on part A?

• What is the probability that the spinner will
  stop on
• An even number?
• An odd number?

3. What fraction names the probability that the
spinner will stop in the area marked A?
21
Probability Activity 2

• In your group, open your MM bag and put the
  candy on the paper plate.
• Put ten brown MMs and five yellow MMs in the
  bag.
• Ask your group, what is the probability of
  getting a brown MM?
• Ask your group, what is the probability of
  getting a yellow MM?

22

• Another person in the group will then put in 8
  green MMs and 2 blue MMs.
• Ask the group to predict which color you are more
  likely to pull out, least likely, unlikely, or
  equally likely to pull out.
• The last person in the group will make up his/her
  own problem with the MMs.

23
Finding Probabilities of Events
You roll a six-sided number cube whose sides are
numbered from 1 through 6.
Find the probability of rolling a 4.
SOLUTION
Only one outcome corresponds to rolling a 4.
number of ways to roll a 4
P (rolling a 4)
24
Finding Probabilities of Events
You roll a six-sided number cube whose sides are
numbered from 1 through 6.
Find the probability of rolling an odd number.
SOLUTION
Three outcomes correspond to rolling an odd
number rolling a 1, 3, or a 5.
P (rolling odd number)
number of ways to roll an odd number
25
Probability of multiple Events

• First you must determine if the events are
  independent or dependent
• Independent events Events that do not affect
  one another.
• Dependent events When the first event affects
  the probability of the other(s).

26
Event 1

• Events that do not have an effect on one another.
  

If we are choosing 2 cards from a deck and after
we choose the first card we replaces it shuffle
and choose again.
Does the first pick affect the second?
No, so this means the 2 events are independent.
27
Event 2

• Events that do not have an effect on one another.
  

If we are choosing 2 cards from a deck and after
we choose the first card we do not replaces it,
we just choose again.
Does the first pick affect the second?
Yes because there are less card now, so this
means the 2 events are dependent.
28
A Deck of cards
29
Are the events Independent or Dependent Events?

• Rolling 3 number cubes
• Independent one number does not affect the
  others
• Choosing 2 marbles from a bag with out
  replacement.
• Dependent each time you choose you have one
  less marble

30
Are the events Independent or Dependent Events?

• Rolling a number cube and flipping a coin
• Independent the number cube does not affect the
  coin

31
Counting outcomes of multiple events

• Making a list
• Tree Diagrams
• Quick multiplication
• Combination and Permutation

32
Making a list
A
B
C
D
E
F
G

• PROBLEM Phillip will shuffle the cards and
  choose three without looking or replacing them.
  How many different combinations are possible?

33
Making a list
A
B
C
D
E
F
G

• When making a list start with the first letter
  and list all possibilities using that letter.

AFB
ABC
ACB
ADB
AEB
AGB
ABC
ACB
AFC
ADC
AEC
AGC
ABD
ACD
AFD
AGD
ABE
ACE
ADE
AED
AFE
ABF
ACF
ADF
AEF
AGE
AFG
ABG
ACG
ADG
AEG
AGF
Does order matter?
No
Can we eliminate any?
Yes
34
Making a list
A
B
C
D
E
F
G

• When making a list start with the first letter
  and list all possibilities using that letter.

ABC
ABD
ACD
ABE
ACE
ADE
ABF
ACF
ADF
AEF
AFG
ABG
ACG
ADG
AEG
How many combinations are left?
15
35
Making a Tree Diagram

• A tree diagram is an image made up of a branching
  structure, which is used to show connections
  between items, topics or ideas.

36
Making a Tree Diagram

• In Probability or Data analysis we use a tree
  diagram to show all possible outcomes for a given
  situation.

37
Making a Tree Diagram

• To make a tree diagram we must begin by
  identifying the first stage or choice.

38
Making a Tree Diagram

• Example
• You sit down at a restaurant for a meal and on
  the menu there are 3 salads 4 main courses and 3
  deserts. How many different 3 course meals are
  possible.

Salads
Main courses
Deserts
39
Salads
Main courses
Deserts
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
1
3
2
3
1
2
1
3
2
3
1
2
3
40
Activity 3 Tree Diagram

• Get into groups of 3
• In each group you are to design a tree diagram on
  your poster paper.
• Make sure to include a key if you are using
  acronyms.
• Activity is also included

41
Activity 4 Permutations

• Sample Situation
• Ms. Jones uses the four letters A, B, C, and
  D in different orders to assign computer log-in
  passwords to her 20 students. Each letter appears
  only once in a password, but the order of the
  letters can be different.

42
Activity 4 Permutations

• 1. Complete the tables by forming four-letter
  passwords.

43
Activity 4 Permutations

1. Are there enough possible passwords for 20
   students?
2. What is the total number of passwords that Ms.
   Jones can create?

44
Activity 4 Permutations

• 4. Discussion Explain how to find the number of
  passwords possible if there are 5 letters
  available, A, B, C, D, and E.

45
Permutation

• Order matters !!!
• The passwords ABCD and ABDC are different because
  order matters
• If the order is of significance, the
  multiplication rules are often used when several
  choices are made from one and the same set of
  objects.

46
Permutations- Definition

• In general, if r objects are selected from a set
  of n objects, any particular arrangement of these
  r objects (say, in a list) is called a
  permutation.
• In other words, a permutation is an ordered
  arrangement of objects.
• By multiple principle, the total number of
  permutations of r objects selected from a set of
  n objects is n(n-1)(n-2)(n-r1)

47
Permutations More examples

• Examples
• How many permutations of 3 of the first 5
  positive integers are there?
• How may permutations of the characters in
  COMPUTER are there? How many of these end in a
  vowel?
• How many batting orders are possible for a
  nine-man baseball team?

48
Permutations - Calculation

• Background-Factorial notation
• 1!1, 2!(2)(1)2, 3!(3)(2)(1)6
• In general, n! n(n-1)(n-2) 321 for any
  positive integer n.
• It is customary to let 0!1 by definition.
• Calculation of Permutation

49
Permutations -- Special Cases

• P(n,0)
• Theres only one ordered arrangement of zero
  objects, the empty set.
• P(n,1)
• There are n ordered arrangements of one object.
• P(n,n)
• There are n! ordered arrangements of n distinct
  objects (multiplication principle)

50
Combinations

• An NBA team has 12 players, in how ways we can
  choose 5 from 12?
• Can we use permutations?
• Are we interested in the order of the players?

51
Combinations (cont.)

• A combination is the same as a subset.
• When we ask for the number of combinations of r
  objects chosen from a set of n objects, we are
  simply asking How many different subsets of r
  objects can be chosen from a set of n objects?
• The order does not matter.

52
Combinations (cont.)

• Any r objects can be arranged among themselves in
  r! permutations, which only count as one
  combination.
• So the n(n-1)(n-2)??(n-r1) different
  permutations of r objects chosen from a set of n
  objects contain each combination r! times.

53
Combinations -- Definition

• The number of combinations of r objects
• chosen from a set of n objects is
• for r0,1,2,,n
• Or
• Other notations for C(n,r) are

54
Combinations (cont.)

• For each combination, there are r! ways to
  permute the r chosen objects.
• Using the multiplication principle
• C(n,r)r!P(n,r)

are refer as binomial coefficients
55
Combinations More examples

• In how many ways a committee of five can be
  selected from among the 80 employees of a
  company?
• In how many ways a research worker can choose
  eight of the 12 largest cities in the United
  States to be included in a survey?

56
Combinations (cont.)

• Lets introduce a simplification
• When we choose r objects from a set of n
  objects we leave (n-r) of the n objects, so there
  are as many ways of leaving (or choosing) (n-r)
  objects as there are of choosing r objects.
• So for the solution of the previous problem, we
  have

57
Combinations -- Special Cases
there is only one way to chose 0 objects from the
n objects

• C(n,0)
• C(n,1)
• C(n,n)

there are n ways to select 1 object from n objects
there is only one way to select n objects from n
objects, and that is to choose all the objects
58
Permutations or Combinations ?

• There are fewer ways in a combinations problem
  than a permutations problem.
• The distinction between permutations and
  combinations lies in whether the objects are to
  be merely selected or both selected and ordered.
  If ordering is important, the problem involves
  permutations if ordering is not important the
  problem involves combinations.

59
Permutations or Combinations ?

• C(n,r) can be used in conjunction with the
  multiplication principle or the addition
  principle.
• Thinking of a sequence of subtasks may seem to
  imply ordering bit it just sets up the levels of
  the decision tree, the basis of the
  multiplication principle.
• Check the Fig 3. 9 to get an idea about the
  difference between permutation and combination.

60
Eliminating duplicate

• A committee of 8 students is to be selected from
  a class consisting of 19 freshmen and 34
  sophomores. In how many ways can a committee with
  at least 1 freshman be selected?

61
Eliminating duplicate

• How many distinct permutations are there of the
  characters in the word Mongooses?
• How many distinct permutations are there of the
  characters in the word APALACHICOLA?

62
Eliminating duplicate (cont.)

• In general, suppose there are n objects of
  which a set of n1 are indistinguishable for each
  other, another set of n2 are indistinguishable
  from each other, and so on, down to nk objects
  that are indistinguishable from each other. The
  number of distinct permutations of the n objects
  is

63
Combinations with Repetitions

• A jeweler designing a pin has decided to use two
  stones chosen from diamonds, rubies and emeralds.
  In how many ways can the stones be selected?
• Answer-- D,R, D,D, D,E, E,R,E,E, R,R.
• Any other way to solve this problem? What if he
  needs five stones?

64
Combinations with Repetitions (cont.)

• Some hints?
• 1 diamond, 3 rubies and 1 emerald
• 5 diamond, 0 rubies and 0 emerald
• 0 diamond, 5 rubies and 0 emerald
• 0 diamond, 0 rubies and 5 emerald
• What is it? Choose 5 stars from 7 elements,
• i.e.C(7,5)

65
Combinations with Repetitions (cont.)

• In general, there must be n-1 markers to indicate
  the number of copies of each of the n objects.
• We will have r (n-1) slots to fill (objects
  markers).
• We want the number of ways to select r out of the
  previous slots to fill.
• Therefore we want
• Six children use one lollipop each from a
  selection of red, yellow, and green lollipops. In
  how many ways can this be done?

66
Activity 4 Find a Sample Space

• The set of all possible outcomes of a
  probability experiment is called the sample
  space. The sample space may be quite small, as it
  is when you toss a coin (sample space heads or
  tails).
• Situation A number cube is numbered from 1 to
  6. A spinner has 5 equal sections lettered from A
  through E. You can use an organized list to find
  the sample space for an experiment.

67
How can you tell which is experimental and which
is theoretical probability?

• Experimental
• You tossed a coin 10 times and recorded a head 3
  times, a tail 7 times
• P(head) 3/10
• P(tail) 7/10
• You actually perform the task

• Theoretical
• Toss a coin and getting a head or a tail is 1/2.
• P(head) 1/2
• P(tail) 1/2
• You dont actually do the task

68
Experimental probability

• Experimental probability is found by repeating an
  experiment and observing the outcomes.

P(head) 3/10 A head shows up 3 times out of 10
trials, P(tail) 7/10 A tail shows up 7 times
out of 10 trials
69
Compare experimental and theoretical probability

• Both probabilities are ratios that compare the
  number of favorable outcomes to the total number
  of possible outcomes

P(head) 3/10 P(tail) 7/10
P(head) 1/2 P(tail) 1/2
70
Contrast Experimental and theoretical probability
71
Identifying the Type of Probability

• A bag contains three red marbles and three blue
  marbles.
• P(red) 3/6 1/2
• Theoretical
• (The result is based on the possible outcomes)

72
Identifying the Type of Probability

• You draw a marble out of the bag, record the
  color, and replace the marble. After 6 draws, you
  record 2 red marbles
• P(red) 2/6 1/3
• Experimental
• (The result is found by repeating an experiment.)

73
How come I never get a theoretical value in both
experiments? Tom asked.

• If you repeat the experiment many times, the
  results will getting closer to the theoretical
  value.
• Law of the Large Numbers

74
Law of the Large Numbers 101

• The Law of Large Numbers was first published in
  1713 by Jocob Bernoulli.
• It is a fundamental concept for probability and
  statistic.
• This Law states that as the number of trials
  increase, the experimental probability will get
  closer and closer to the theoretical probability.
• http//en.wikipedia.org/wiki/Law_of_large_numbers

75
Contrast Experimental and theoretical probability

• Three students tossed a coin 50 times
  individually.
• Lisa had a head 20 times. ( 20/50 0.4)
• Tom had a head 26 times. ( 26/50 0.52)
• Al had a head 28 times. (28/50 0.56)
• Please compare their results with the theoretical
  probability.
• It should be 25 heads. (25/50 0.5)

76
Large Number activity

• Each person is to roll a number cube 20 time and
  record their results in the following table.

Number rolled Theoretical Probability Frequency Experimental Probability
1
2
3
4
5
6
77
Activity 6 Rock Paper Scissors

• Place the class into groups and use the activity
  included.
• Activity PDF

78
GEOMETRIC PROBABILITY
Some probabilities are found by calculating a
ratio oftwo lengths, areas, or volumes. Such
probabilities arecalled geometric probabilities.
79
Using Area to Find Probability
You throw a dart at the board shown. Your dart is
equally likely to hit any point inside the square
board. Are you more likely to get 10 points or 0
points?
80
Using Area to Find Probability
Are you more likely to get 10 points or 0 points?
SOLUTION
area of smallest circle
P (10 points)
area outside largest circle
P (0 points)
You are more likely to get 0 points.
81
Monte Carlo Area Activity

• Materials
• Blow-up globe.
• Paper and pencil
• Table on slide 75

82
Monte Carlo Area Activity

• Throw the globe to a participant and have them
  catch it with just their finger tips.
• The participant counts how many are touching land
  and how many are not.
• Enter the information onto the table on the next
  slide

83
Monte Carlo Area Activity
Trial of fingers on land for that trial Experimental Probability Cumulative of fingers on land Cumulative Experimental Probability
1
2
3
4
5
6
7
8
9
84
Monte Carlo Area Activity

• Experimental probability
• Total surface area of the Earth
• Approximately 510,065,600 km2
• Multiply the 2 and you should get an
  approximation of the surface area of land on
  Earth.
• Approximately 148,939,100 km2

85
Lesson Review

• Probability as a measure of likelihood
• There are two types of probability
• Theoretical--- theoretical measurement and can be
  found without experiment
• Experimental--- measurement of a actual
  experiment and can be found by recording
  experiment outcomes
• Please click here to take the quiz

86
Lesson Review

• 4 ways for counting outcomes
• Making a list
• Tree Diagram
• Combination
• Permutation
• Geometric Probability

87
Probability Questions

• 1) Lawrence is the captain of his track team.
  The team is deciding on a color and all eight
  members wrote their choice down on equal size
  cards. If Lawrence picks one card at random,
  what is the probability that he will pick blue?

blue
blue
green
black
yellow
blue
black
red
88

• 2) Donald is rolling a number cube labeled 1 to
  6. Which of the following is LEAST LIKELY?
• an even number
• an odd number
• a number greater than 5

89
1
2
3. What is the chance of spinning a number
greater than 1?
4
3
4. What is the chance of spinning a 4? 5. What is
the chance that the spinner will stop on an odd
number?
4
1
2
3
5
6. What is the chance of rolling an even number
with one toss of on number cube?