probability and statistics teacher quality grant

Probability and Statistics

Teacher Quality Grant

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

Key Words

• Experimental probability• Theoretical probability• Law of Large Numbers• Outcome• Event• Random

Click here to check the words

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)

Probability

P=0 P=1P=

€

1

2Event will never occur

Event is equally likely to occur or not occur

Event will always occur

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.

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%.

• 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%.


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


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.


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

THEORETICAL PROBABILITY

THE THEORETICAL PROBABILITY OF AN EVENT

When all outcomes are equally likely, thetheoretical probability that an event A will occur is:

P (A) = total number of outcomes

The theoretical probability of an event is often simply called the probability of the event.

all possible outcomes

number of outcomes in A

outcomes in event A

outcomes in event A

You can express a probability as a fraction, a decimal, or a percent.

For example: , 0.5, or 50%.12

P (A) = 49

Theoretical probability

P(head) = 1/2P(tail) = 1/2Since there are only

two outcomes, you have 50/50 chance to get a head or a tail.

HEADS

TAILS

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

2. What is the probability that the spinner will stop on

(a) An even number?(b) An odd number?

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

AB

C D

3 1

2

A

C B

Theoretical probability


• In your group, open your M&M bag and put the candy on the paper plate.

• Put ten brown M&Ms and five yellow M&Ms in the bag.

• Ask your group, what is the probability of getting a brown M&M?

• Ask your group, what is the probability of getting a yellow M&M?

• Another person in the group will then put in 8 green M&Ms and 2 blue M&Ms.

• 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 M&Ms.

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.

SOLUTIONOnly one outcome corresponds to rolling a 4.

P (rolling a 4) = number of ways to roll a 4

number of ways to roll the die

1

6=

Finding Probabilities of Events

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 numbernumber of ways to roll the die

You roll a six-sided number cube whose sides are numbered

from 1 through 6.

Find the probability of rolling an odd number.

SOLUTION

3

6

1

2= =

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).

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.

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.

A Deck of cards

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

Are the events Independent or Dependent Events?

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

coin

Counting outcomes of multiple events

• Making a list

• Tree Diagrams

• Quick multiplication

• Combination and Permutation

Making a list

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

AA BB CC DD EE FF GG

Making a list

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


ABC

ABD

ABE

ABF

ABG

ACB

ACD

ACE

ACF

ACG

ADB

ADC

ADE

ADF

ADG

AEB

AEC

AED

AEF

AEG

AFB

AFC

AFD

AFE

AFG

AGB

AGC

AGD

AGE

AGF

Does order matter?Can we eliminate any?

No

Yes

ABC ACB

Making a list

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


ABC

ABD

ABE

ABF

ABG

ACD

ACE

ACF

ACG

ADE

ADF

ADG

AEF

AEG AFG

How many combinations are left? 15

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.


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


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


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

Salads Main courses Deserts

1

2

3

A

B

C

D

A

B

C

D

A

B

C

D

1

32

1

32

1

32

1

32

1

32

1

32

1

32

1

321

32

1

32

1

32

1

32

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

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.


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


2. Are there enough possible passwords for 20 students?

3. What is the total number of passwords that Ms. Jones can create?


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

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.

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-r+1)

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?

Permutations - Calculation• Background-Factorial notation:

– 1!=1, 2!=(2)(1)=2, 3!=(3)(2)(1)=6– In general, n!= n(n-1)(n-2) ·…·3·2·1 for any positive

integer n.– It is customary to let 0!=1 by definition.

• Calculation of Permutation)1()2)(1( rnnnnPrn

)!(

)!)(1()2)(1(

rn

rnrnnnn

)!(

!

rn

n

Permutations -- Special Cases1. P(n,0)

There’s only one ordered arrangement of zero objects, the empty set.

2. P(n,1)

There are n ordered arrangements of one object.3. P(n,n)

There are n! ordered arrangements of n distinct objects (multiplication principle)

1!

!

)!0(

!)0,(

n

n

n

nnP

nn

nnP

)!1(

!)1,(

!!0

!

)!(

!),( n

n

nn

nnnP

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?

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.


• 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-r+1) different permutations of r objects chosen from a set of n objects contain each combination r! times.

Combinations -- Definition

The number of combinations of r objects chosen from a set of n objects is:

for r=0,1,2,…,n

Or

Other notations for C(n,r) are:

!

)1()2)(1(

r

rnnnn

!)!(

!

rrn

n

r

nCC nrrn ,,

• For each combination, there are r! ways to permute the r chosen objects.

• Using the multiplication principle:C(n,r)r!=P(n,r)

(Number of ways to choose the objects)

(Number of ways to arrange the objects chosen)*

r

nare refer as binomial coefficients


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?

016,040,24!5

7677787980

5

80

495320,4024

600,001,479

!8!4

!12

8

12

Combinations –More examples

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:

rn

n

r

n

495!4

9101112

4

12

8

12


Combinations -- Special Cases

• C(n,0):

• C(n,1):

• C(n,n):

1)!0(!0

!)0,(

n

nnC

nn

nnC

)!1(!1

!)1,(

1)!(!

!)0,(

nnn

nnC

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

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.

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.

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?

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?

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

)!.().........!)(!(

!

21 knnn

n

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?

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)

• 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?

)!1(!

)!1(

)!1(!

)!1(),1(

nr

nr

rnrr

nrrnrC

Combinations with Repetitions (cont.)

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.

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)= 3/10P(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 don’t actually do

the task

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

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

Contrast Experimental and theoretical probability

Experimental Vs. Theoretical

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Exp P(H) P(H) Exp P(T) P(T)

Lisa

Tom

Al

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)

Identifying the Type of Probability

Trial Red Blue

1 1

2 1

3 1

4 1

5 1

6 1

Total 2 4

Exp. Prob. 1/3 2/3

• 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.)

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

Experimental VS. Theoretical

50

53.4

48.948.4

49.87

45

46

47

48

49

50

51

52

53

54

1

Thoeretical5-trial10-trial20-trial30-trial

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

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)

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

Activity 6: Rock Paper Scissors

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

GEOMETRIC PROBABILITY

Some probabilities are found by calculating a ratio oftwo lengths, areas, or volumes. Such probabilities arecalled geometric probabilities.

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?

•32

182= = = ≈ 0.0873

3249

36

Using Area to Find Probability

SOLUTION

P (10 points) =area of smallest circle

area of entire board

Are you more likely to get 10 points or 0 points?

You are more likely to get 0 points.

P (0 points) =area outside largest circle

area of entire board

182 – ( • 9 2 )

182= = = ≈ 0.215

324

324 – 814

4 –

Monte Carlo Area Activity

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


• 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

Monte Carlo Area ActivityTrial # # 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


• 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

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

Lesson Review• 4 ways for counting outcomes

– Making a list

– Tree Diagram

– Combination

– Permutation

• Geometric Probability

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?

yellow

red

blue blue

blue

green black

black

2) Donald is rolling a number cube labeled 1 to 6. Which of the following is LEAST LIKELY?

A. an even numberB. an odd numberC. a number greater than 5

3. What is the chance of spinning a number greater than 1?

4. What is the chance of spinning a 4?5. What is the chance that the spinner will

stop on an odd number?

6. What is the chance of rolling an even number with one toss of on number cube?

1 2

4 3

4 1

2 35

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