Title: Probability and Statistics
1Probability and Statistics
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8What 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
9Key Words
- Experimental probability
- Theoretical probability
- Law of Large Numbers
- Outcome
- Event
- Random
- Click here to check the words
10Definition 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)
11Probability
P0
P1
Event will never occur
Event is equally likely to occur or not occur
Event will always occur
12Chance / 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.
13Probability 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.
14Probability 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.
15Probability 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
16Probability 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.
17Experimental 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
18THEORETICAL 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
19Theoretical 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.
20Theoretical 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?
21Probability 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.
23Finding 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)
24Finding 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
25Probability 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).
26Event 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.
27Event 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.
28A Deck of cards
29Are 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
30Are the events Independent or Dependent Events?
- Rolling a number cube and flipping a coin
- Independent the number cube does not affect the
coin
31Counting outcomes of multiple events
- Making a list
- Tree Diagrams
- Quick multiplication
- Combination and Permutation
32Making 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?
33Making 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
34Making 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
35Making 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.
36Making a Tree Diagram
- In Probability or Data analysis we use a tree
diagram to show all possible outcomes for a given
situation.
37Making a Tree Diagram
- To make a tree diagram we must begin by
identifying the first stage or choice.
38Making 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
39Salads
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
40Activity 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
41Activity 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.
42Activity 4 Permutations
- 1. Complete the tables by forming four-letter
passwords.
43Activity 4 Permutations
- Are there enough possible passwords for 20
students? - What is the total number of passwords that Ms.
Jones can create?
44Activity 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.
45Permutation
- 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.
46Permutations- 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)
47Permutations 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?
48Permutations - 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
49Permutations -- 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)
50Combinations
- 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?
51Combinations (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.
52Combinations (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.
53Combinations -- 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
57Combinations -- Special Cases
there is only one way to chose 0 objects from the
n objects
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
58Permutations 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.
59Permutations 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.
60Eliminating 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?
61Eliminating 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?
62Eliminating 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
63Combinations 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?
64Combinations 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)
65Combinations 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?
66Activity 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.
67How 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
68Experimental 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
69Compare 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
70Contrast Experimental and theoretical probability
71Identifying 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)
72Identifying 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.)
73How 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
-
74Law 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
75Contrast 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)
76Large 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
77Activity 6 Rock Paper Scissors
- Place the class into groups and use the activity
included. - Activity PDF
78GEOMETRIC PROBABILITY
Some probabilities are found by calculating a
ratio oftwo lengths, areas, or volumes. Such
probabilities arecalled geometric probabilities.
79Using 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?
80Using 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.
81Monte Carlo Area Activity
- Materials
- Blow-up globe.
- Paper and pencil
- Table on slide 75
82Monte 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
83Monte 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
84Monte 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
85Lesson 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
86Lesson Review
- 4 ways for counting outcomes
- Making a list
- Tree Diagram
- Combination
- Permutation
- Geometric Probability
87Probability 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?