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P1-IntroProbability.pdf

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LSP 121

P1 - Introduction to Probability

Probability Outline • Concepts/Vocabulary • Sample space and sample space size • Calculating Probability • Types of Probability

– theoretical – empirical

• Theoretical vs. Empirical probability • Other useful probability calculations • Gambler's Fallacy • Expected Value • Type I/Type II error

Probability

• How likely is it that something will happen ?

Expressing Probability

Event

• Something that occurs

• For example: roll of dice, flip of coin, weather forecast, election

Outcome

• Result of an event

• Has a value of interest to us

• For example: value on rolled die is 6, heads, rain, Bob Smith is elected treasurer

Sample Space

• List of all possible outcomes of an event or multiple events

Sample Space =

All Possible Outcomes

Sample Space = All Possible Outcomes

• Single event

• Examples Flip one coin

sample space = Tail ,Head

Roll one die sample space = 1,2,3,4,5,6

Sample Space = All Possible Outcomes

• Multiple events

• Example - Flip two coins

Sample space all combinations Head,Head Head,Tail Tail,Head Tail,Tail

Sample Space Size

Number of (Count of ) possible outcomes in the sample space for a single event

Number of (Count of) all possible combinations in the sample space for multiple events

Sample Space Size

Sample Space Size =

Number of Possible Outcomes =

Number of Possible Combinations

Sample Space Size for Single Event

• number of the possible outcomes – count them

• Examples Flip one coin

sample space size = 2

Roll one die sample space size = 6

Sample Space Size for Multiple Events

multiply the sample space size of each of the events

for example: Sample space size for throwing two dice =

sample space size for one die x

sample space size for second die

Sample Space Size = Number of Possible Outcomes for multiple events

M = number of possible outcomes for event A N = number of possible outcomes for event B O = number of possible outcomes for event C

Total number of possible outcomes (size of sample space ) for events A and B and C (the three events happen together)

M x N x O

Sample Space Size for Two Dice

Sample space size for throwing two dice = sample space size for one die = 6

x sample space size for second die = 6

= ?

In this case, we can visualize all outcomes….

Possible Combinations/Sample Space and Values when throwing Two Dice

Die 1

D ie

Value Die 1

V a lu

e D

ie 2

1 2 3 4 5 6

1 2 3 4 5 6 7

2 3 4 5 6 7 8

3 4 5 6 7 8 9

4 5 6 7 8 9 10

5 6 7 8 9 10 11

6 7 8 9 10 11 12

Combined value of both dice 17

Calculating Sample Space Size – Example #1

• A restaurant menu offers – two choices for an appetizer, – five choices for a main course, and

– three choices for a dessert.

How many different possible three-course meals are there?

Calculate Sample Space Size – Example #1

• A restaurant menu offers two choices for an appetizer, five choices for a main course, and three choices for a dessert.

How many different possible three-course meals are there?

2 x 5 x 3 = 30

Calculating Sample Space Size – Example #2

• During one quarter, a college offers – 3 natural science classes, – 4 social science classes, – 10 English classes, and

– 2 fine arts classes.

How many possible four-class combinations are there ?

Calculating Sample Space Size – Example #2

• During one quarter, a college offers – 3 natural science classes, – 4 social science classes, – 10 English classes, and

– 2 fine arts classes.

How many possible four-class combinations are there ?

3 x 4 x 10 x 2 = 240

To calculate Number of Possible Combinations

(Sample Space Size)

1. Draw a diagram of the components 2. Note the number of values for each

component 3. Multiply the numbers of values together

The result is the number of possible combinations.

Example – Number of Possible Combinations

A bank allows you to select an identification code that is five places long.

There are limits for the values that can be in each of the five places:

1. the first place must be an upper-case or lower-case letter in the English alphabet

2. the second place must be a lower-case letter in the English alphabet

3. the third place must be a number 0-9 4. the fourth place must be an upper-case letter in the English

alphabet or a lower-case letter in the English alphabet or a number 0-9

5. the fifth place must be Y or N How many possible 5-place combinations are there ?

Number of combinations – approach (cont.)

• write the places in the code as a series of X's

X X X X X

• write the number of possible values below each X

X X X X X 26+26 26 10 26+26+10 2

Number of combinations – approach (cont.)

• Calculate the number of possible values below each X

X X X X X 26+26 26 10 26+26+10 2

52 26 10 62 2

Number of combinations – approach (cont.)

• Multiply the numbers of possible values together

52 x 26 x 10 x 62 x 2

Number of possible combinations = 1,676,480

Among other uses….

• Number of possible combinations (sample space size) can be used in calculating probabilities

In general

Probability of an Outcome Occurring

P(A) = proportion of the event(s) in which a particular outcome (A) occurs

Calculating Probability

For an event:

Probability =

number of outcomes of a particular value ______________________________________________________________________

number of possible outcomes (all possible combinations)

Calculating Probability of an Outcome Occurring

• For example, – Rolling a die (event) has 6 possible outcomes,

(1,2,3,4,5,6).

– Each one of those possible outcomes can occur one way

– Probability of any of those outcomes (e.g. rolling a 3) is 1/6

Expressing Probability

• As a proportion 0.0 � 1.0

• As a percentage 0 � 100 %

• As a fraction e.g. ¼

• As a number of outcomes e.g. 1 in 6

Types of Probability

Basic Types of Probability

• Theoretical, or a priori

• Empirical

Theoretical or a Priori

• Theoretical, or a priori probability –

– Can be calculated before event

Empirical Probability

– based on the results of observations or experiments.

Theoretical Probability

Calculated before an event occurs

Calculating Probability

Theoretical Probability =

number of ways an outcome can occur ______________________________________________________________________

number of possible outcomes (sample space size)

Theoretical Probability – calculated before an event

• P(A) = (number of ways A can occur) _______________________________________________________________

(total number of outcomes (sample space size))

e.g. #1 Probability of a head landing in a coin toss

#2 Probability of rolling a 5 with one die

#3 Probability that your phone number ends with 9

Theoretical Probability Example #1

Theoretical Probability of a head landing in a coin toss

number of ways "heads" can occur = 1

sample space size (heads, tails) = 2

theoretical probability of landing heads in a coin toss =

½ or .5 or 50 % or 1 of 2

Theoretical Probability Example #2

Theoretical Probability of rolling a "5" with one die

number of ways "5" can occur = 1

sample space size (1,2,3,4,5,6) = 6

theoretical probability of rolling a "5" with one die =

1/6 or .1667 or 16.67 % or 1 of 6

Theoretical Probability Example #3

Theoretical Probability your phone number ends with "9"

number of ways "9" can occur = 1

sample space size (0,1,2,3,4,5,6,7,8,9) = 10

theoretical probability your phone number ends with "9" =

1/10 or .10 or 10.0 % or 1 of 10

Calculating Theoretical Probabilities "Either/Or"

• What is the theoretical probability of either "this" outcome happening or "that" outcome happening ?

• Assume that the outcomes don’t overlap

Calculating Theoretical Probabilities "Either/Or"

• What is the theoretical probability of either "this" outcome happening or "that" outcome happening ?

Add the theoretical probabilities of each P(A or B) = P(A) + P(B)

Calculating Theoretical Probabilities "Either/Or"

Example: You roll a single die.

What is the probability of rolling either a 2 or a 3?

Calculating Theoretical Probabilities "Either/Or"

Example: You roll a single die. What is the probability of rolling either a 2 or a 3?

P(2 or 3) = P(2) + P(3) =

1/6 + 1/6 = 2/6 = 1/3

Calculating Theoretical Probabilities - Multiple Independent Events

"And" • Event A and Event B and Event C, etc. all

occur

• Assume all events are independent – the outcome of one does not affect the

outcome of the other

Calculating Theoretical Probabilities - Multiple Independent Events

"And"

• The probability of all events occurring, multiply the probabilities

P(A and B and C), = P(A) x P(B) x P(C)

Calculating Theoretical Probabilities Multiple Independent Events

"And"

Example #1: What is the theoretical probability that both your phone number and mine end in "8" ?

Calculating Theoretical Probabilities Multiple Independent Events

"And"

Example #1: What is the theoretical probability that both your phone number and mine end in "8" ?

Theoretical Probability that your phone number ends in "8" = .1 or 1/10

Theoretical Probability that my phone number ends in "8" = .1 or 1/10

Calculating Theoretical Probabilities Multiple Independent Events

"And"

Example #1: What is the theoretical probability that both your phone number and mine end in "8" ?

Theoretical Probability that your phone number ends in "8" =.1

Theoretical Probability that my phone number ends in "8" = .1

Theoretical Probability that both of our phone numbers end in "8" =

.1 x .1 = .01 or 1/10 x 1/10 = 1/100

Calculating Theoretical Probabilities Multiple Independent Events

"And" Example #2

• You toss three coins. What is the theoretical probability of getting three tails?

That is…. Coin Toss Result for Coin A = Tails

and

Coin Toss Result for Coin B = Tails and

Coin Toss Result for Coin C = Tails

Calculating Theoretical Probabilities Multiple Independent Events

“And” • Example #2 • You toss three coins.

– What is the theoretical probability of getting three tails?

1/2 x 1/2 x 1/2 = 1/8

Coin A Coin B Coin C

Calculating Theoretical Probabilities - Multiple Independent Events

“And”

Example #3

• Find the theoretical probability that a 100-year flood will strike a city in two consecutive years

(FEMA definition of a 100-year flood is that it occurs in 1% of the years in question.)

Calculating Theoretical Probabilities - Multiple Independent Events

“And”

Example #3

• Find the theoretical probability that a 100-year flood will strike a city in two consecutive years

(1 in 100) x (1 in 100) = Year 1 Year 2

Calculating Theoretical Probabilities - Multiple Independent Events

“And” Example #3

• Find the theoretical probability that a 100-year flood will strike a city in two consecutive years

(1 in 100) x (1 in 100) = 1/10000

(1 in 100) x (1 in 100) = 0.01 x 0.01 = 0.0001

Year 1 Year 2

Empirical Probability

– Calculated based on the results of observations or experiments.

Empirical Probability – calculated after events occur

• Empirical P(A) =

(number of A outcomes) _______________________________________________________________

(total number of observations)

Empirical Probability – calculated after events

Example #1

– Weatherperson Fred has predicted the weather 1000 times

– Weatherperson Fred has predicted the weather correctly 250 times

Empirical Probability – calculated after events

Example #1

– Weatherperson Fred has predicted the weather 1000 times

– Weatherperson Fred has predicted the weather correctly 250 times

– The empirical probability of Weatherperson Fred predicting the weather correctly is 250/1000 = 1/4

Empirical Probability Example #2

• Records indicate that the Funky River has crested above flood level just four times in the past 2000 years.

– What is the empirical probability that the Funky River will crest above flood level in any given year? (measurement is whether a flood has occurred in a particular year, yes or no)

Empirical Probability Example #1

• Records indicate that the Funky River has crested above flood level just four times in the past 2000 years.

– What is the empirical probability that the Funky River will crest above flood level in any given year?

4/2000 = 1/500 = 0.002

Comparing Theoretical and Empirical Probabilities

• Theoretical probability of a coin flip resulting in heads = .5

• The empirical probability (your actual results) when flipping a coin may not be the same

Are theoretical and empirical probabilities equal for an

outcome ?

Create a probability matrix to display and compare the theoretical and empirical

probabilities

Probability Matrix

Outcomes

(Sample Space)

Theoretical Probability

Number of times this combination appears in this experiment

Empirical Probability

Compare Theoretical and Empirical Probability Example

• Flip two coins 15 times. – What is the theoretical probability of each

outcome ?

– What is the empirical probability of each outcome ?

What is the sample space for two coins?

Tossing 2 coins has 4 possible outcomes – Sample space is HT,HH,TH,TT

Enter all the possible outcomes in the left hand column (outcomes) of the probability matrix

Update Probability Matrix with all possible outcomes

Outcomes

(Sample Space)

Theoretical Probability

Number of times this combination appears in this experiment

Empirical Probability

Total 1.0 1.0 68

What is the sample space size for flipping two coins?

Sample space size for tossing 2 coins is 4

(sample space size for coin A) x (sample space size for coin B )

2 x 2 = 4

There should be 4 outcomes in the Outcomes column.

Calculate Theoretical Probability

• Theoretical probability =

ways outcome can occur = 1 sample space size = 4

in this case theoretical probability for each outcome is ¼

Add Theoretical Probability to the matrix

• Theoretical probability =

ways outcome can occur = 1 sample space size = 4

Total theoretical probability should equal 1.

Add Theoretical Probabilties

Outcomes

(Sample Space)

Theoretical Probability

Number of times this combination appears in this experiment

Empirical Probability

HH 1/4

HT 1/4

TH 1/4

TT 1/4

Total 4/4 = 1 72

Calculate Empirical Probabilities

• Empirical probability =

count of each actual outcome total number of events/observations

empirical probability for each outcome depends on the actual results

For this example

• Count for each actual outcome

HH = 2 HT = 5 TH = 4 TT = 4

• Add number of times each outcome actually occurs to the matrix

Add number of occurrences

Outcomes

(Sample Space)

Theoretical Probability

Number of times this combination appears in this experiment

Empirical Probability

HH 1/4 2

HT 1/4 5

TH 1/4 4

TT 1/4 4

Total 1 15 75

Total of actual occurrences should equal number of trials

• Count for each actual outcome

HH = 2 HT = 5 TH = 4 TT = 4

Total = 15 • Add the total number of occurrences to the

matrix – double check 76

Add the total of occurrences Double check that it matches the number of trials

Outcomes

(Sample Space)

Theoretical Probability

Number of times this combination appears in this experiment

Empirical Probability

HH 1/4 2

HT 1/4 5

TH 1/4 4

TT 1/4 4

Total 1 15 77

Calculate Empirical Probabilities

• Empirical probability = count for each actual outcome

15 Empirical probability of HH = 2/15 Empirical probability of HT = 5/15 Empirical probability of TH = 4/15 Empirical probability of TT = 4/15

Total of Empirical Probabilities

• Should be equal to 1

Empirical probability of HH = 2/15 Empirical probability of HT = 5/15 Empirical probability of TH = 4/15 Empirical probability of TT = 4/15

Total = 15/15 = 1 79

Add Empirical Probabilties to the matrix

Outcomes

(Sample Space)

Theoretical Probability

Number of times this combination appears in this experiment

Empirical Probability

HH 1/4 2 2/15

HT 1/4 5 5/15

TH 1/4 4 4/15

TT 1/4 4 4/15

Total 1 15 15/15 = 1 80

Probability Matrix Complete

Outcomes

(Sample Space)

Theoretical Probability

Number of times this combination appears in this experiment

Empirical Probability

HH 1/4 2 2/15

HT 1/4 5 5/15

TH 1/4 4 4/15

TT 1/4 4 4/15

Total 1 15 1 81

Theoretical = Empirical Probability

Compare Probabilities for Each Combination

In this case….

Will theoretical and empirical probability ever be equal ?

• The Law of Large Numbers says that – as you repeat the event (millions of times), – the empirical probability (actual results) will

approach the theoretical probability.

Law of Large Numbers

• http://bcs.whfreeman.com/ips4e/cat_010/a pplets/expectedvalue.html

Thought Question #1:

What would the sample space size be

if we were flipping three coins ?

Thought Question #2:

What would the sample space be

if we were flipping three coins ?

Thought Question #3:

What would the theoretical probabilities be

if we were flipping three coins ?

Other Useful Probability Calculations

Probability of an Event Not Occurring

• P(not A) = 1 - P(A)

• For example, – If the probability of rolling a 3 with one die is

1/6, – then the probability of NOT rolling a 3 with

one die is • 1- 1/6 = 5/6

Probability of At Least Once

• What is the probability of an outcome happening at least once in a number of trials?

• P(at least one outcome A in n trials) =

1 - [P(not A in one trial)]n

Probability of At Least Once Example #1

• What is the probability that a region will experience at least one 100-year flood during the next 15 years?

Probability of At Least Once Example #1

• What is the probability that a region will experience at least one 100-year flood during the next 15 years?

• Probability of a flood in any one year = 1/100 = .01 • Probability of no flood in any one year = 99/100 = .99 • The number of trials = 15

• P(at least one 100-year flood in 15 years) = 1 – (0.99)15 = 1 - 0.860058 = 0.139942

Stated as percentage = 13.9942% 93

Probability of At Least Once Example #2

• You purchase 10 lottery tickets, for which the theoretical probability of winning some prize on a single ticket is 1 in 10 (or .1).

• What is the theoretical probability that you will have at least one winning ticket?

Probability of At Least Once Example #2

• You purchase 10 lottery tickets, for which the probability of winning some prize on a single ticket is 1 in 10 (or .1).

• What is the probability that you will have at least one winning ticket?

• Probability of winning with any one ticket = .1 • Probability of not winning with any one ticket = .9 • Number of trials = 10

P(at least one winner in 10 tickets) = 1 – (0.9)10 = 1 - 0.348678 = 0.651322

Stated as percentage to three decimal places= 65.132%

Calculating Power of

• Excel has a function called Power

Calculating Power of a number with Excel

The "at least once" approach

• use this approach only when the problem uses the phrase "at least once" or "at least one"

Probability of Exactly Once