probability

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1. Sample Space Size / All Possible Combinations

Make sure that you state your answer in a way that does not lose precision. Show all digits.

a. The Personal Identification Number (PIN) for your ATM card is three places long. Any number is permitted in each place. How many possible values of a PIN are there?

b. Your bank has decided to allow English alphabetic characters (upper and lower case are permitted) as well as any numeric digit in your PIN. How many possible PINs are there when that change is made?

c. In addition to b. above, your bank has decided to require that the first place of the PIN be an English upper case alphabetic character. How many possible PINs are there when that additional change is made?

d. In addition to b. and c. above, your bank has changed the PIN size to four places. How many possible PINs are there when that change is made?

2. Theoretical Probability – show each as a fraction

a. What is the theoretical probability of rolling one die and getting a 2 ?

b. What is the theoretical probability of rolling one die and getting a 4 or a 5?

c. What is the theoretical probability of rolling one die and not getting a 5?

d. What is the theoretical probability of rolling one die and getting a 7 ?

3. Theoretical Probability (and)

What is the theoretical probability that your five best friends all have telephone numbers ending in 5? (Hint: first determine the probability of a single telephone number ending with a 5. Then calculate the probability of the five independent events occurring together.) Show your answer as a fraction.

4. Empirical Probability

After recording the forecasts of your local weather predictor for 125 days, you conclude that he/she gave a correct forecast 84 times. Calculate the empirical probability of correct forecasts during that time. Show your answer as a fraction.

5. Probability calculations

Studies of Florida weather show that, historically, the Miami region is hit by a hurricane every 25 years. Calculate the following probabilities based on the historical record.

a. What is the probability that Miami will be hit by a hurricane in any given year? Show as a fraction.

b. What is the probability that Miami will be hit by hurricanes in both of the next two consecutive years? Show as a fraction.

c. What is the probability that Miami will be hit by hurricanes in either the next year or the year after? Show as a fraction.

d. What is the probability that Miami will be hit by at least one hurricane in the next seven years? Show as a percentage with three decimal places

6. Probability calculations – exactly one occurrence

You are looking for a job. Eventually you get six interviews. In each case you are one of four final candidates for the respective job. Let us assume you have a .25 probability of getting each job and the probability of getting each job offer is independent of getting any other job offer.

a.. What is the probability that you get exactly one job offer of the six opportunities?

Show this as a percentage with three decimal places

7. Expected Value – seller

An insurance company sells a policy for $1500.

Based on past data,

an average of 1 in 100 policyholders will win a $25,000 claim on a policy,

an average of 1 in 200 policyholders will win a $50,000 claim on a policy, and

an average of 1 in 500 policyholders will win a $250,000 claim on a policy.

a. Find the expected value (to the company) per policy sold.

b. If the company sells 10,000 policies, what is the expected total profit or loss for the company?

8. Expected Value – buyer

You can buy a lottery ticket for $500.

The odds are as follows:

1 in 10 tickets will pay $500.

1 in 100 tickets will pay $1000.

1 in 1000 tickets will pay $200,000.

1 in 10,000 tickets will pay $1,000,000.

1 in 100,000 tickets will pay $10,000,000.

a. What is the expected value of the ticket to you?

b. Should you buy the ticket? Explain your decision.

9. Type I and Type II Errors

The information for a particular area and radon testing is as follows:

3 of 100 homes have radon gas problems

A test for radon gas is 85% accurate

The test is performed in 10,000 homes in this area

a. Using the table below, create a matrix similar to the one in the Type I-Type II error notes.

Label the cells as true positives, false positives, false negatives, true negatives and Type I and Type II errors.

Complete the matrix to show the values for true positives, false positives (Type I), false negatives (Type II), and true negatives

Calculate the totals and add them to the matrix.

b. What is the chance that a positive radon test really means that there is radon in that house? Show this as a percentage with 3 decimal places.

c. Assume that the test results were negative for a particular house. What is the chance that there really is a radon problem in that house? Show this as a percentage with 3 decimal places.

2 of 4

LSP 1

P2 – Type I and Type II Errors

Test

results

• medical tests

• diagnostic tests

• presence/absence of a condition

Test Results

• Positive – condition is present
– Not a value judgment

• Negative – condition is not present
– Not a value judgment

Take action….

• based on test results

• concerns about taking action
– side effects
– cost
– many others

No test is

0% accurate

Our Concerns

• If test results say condition exists,
(test result is positive)

• does it actually exist ?

• If test results say condition does not exist,
(test result is negative)

• does it actually exist ?

How likely is it that
(given the test results)

a condition exists ?

…that we need to take action

probability

Four Possible Outcomes
(test results vs. reality)

• True Positive
– test results are correct
– condition exists

• True Negative
– test results are correct
– condition does not exist

• False Positive
– test results are not correct
– condition does not exist

• False Negative
– test results are not correct
– condition exists

Test Results vs. Reality

Reality �

Test Results

Has Condition Does Not Have
Condition

Positive Test True
Positive

False
Positive
Type I

Error

Negative Test False
Negative

Type II Error

True
Negative

Technique for
Determining Probability

of each of the four possible
outcomes

Three important factors
stated in each problem

• Test accuracy

•

Occurrence in the real world

•

Number of test participants

Test Accuracy

• What percent of the time does a test
correctly identify the condition ?

• Example:
A disease detection screening is 85%
accurate

Occurrence in the real world

• How frequently does the condition occur in
the population ?

• This could be stated is a variety of ways:
• percentage,
• proportion,
• x of y

• Example:
80% of homes have termites
.80 homes
8 of 10 homes

Number of test participants

• Number of tests run

• Example:
– 10,000 people were tested for high blood

sugar

Example

• We conduct a study in which diagnostic
tests are given to 10,000 people who have
symptoms of Condition D.

• Assume that 1% of people who have
symptoms of Condition D actually have it.

• The test for Condition D is 85% accurate

How many people we
test for Condition D

are in each category ?

• True positive
• False positive (Type I Error)
• True negative
• False negative (Type II Error)

Step 0. Draw basic 4 x 4 matrix
and label cells

Has
Condition D

Does not have
Condition D

Total

Positive
Test

True Positive

Type I Error

Negative
Test

Type II

Error

True Negative

Total

Number of test participants
• We conduct a study in which diagnostic
tests are given to 10,000 people who have
symptoms of Condition D.

Step 1. Post total participants

Has
Condition D

Does not
have
Condition D

Total
Positive
Test
True Positive Type I Error
Negative
Test

Type II Error True Negative

Total

10,000

Occurrence in the Population

• 1% of people who have symptoms of
Condition D actually have it.

Step 2. calculate # in population who do and
do not have the condition, based on

occurrence in the population
Has
Condition D

Does not have
Condition D

Total
Positive
Test
True Positive Type I Error
Negative
Test
Type II Error True Negative

Total 100

99% of 10,000

10,000

1% in the population have Condition D

99% do not 21

1% of 10,000

9,900

Test Accuracy
• The test for Condition D is

85% accurate

Step 3. Calculate inner cell
values based on test accuracy

Has
Condition D

Does not have
Condition D
Total
Positive
Test

True Positive

(85% of 100)
85

Type I Error

(15% of 9900)

1485

Negative
Test
Type II Error

(15% of 100)

True Negative

(85% of 9900)
8415

Total
100 9,900 10,000

85% accurate
85% accurate

Do not round totals in the
interior cells of the matrix

• if the calculation of the values in the four
quadrants result in a number that is not a
whole number (e.g. 15 x .10 = 1.5)

• do not round those values

Complete the matrix

calculate
total positive and total negative

results

Step 4. Calculate
Total Positive and Negative Test Results

Has Condition D Does not have
Condition D

Total
Positive
Test
True Positive
(85% of 100)
85
Type I Error
(15% of 9900)
1485

(85+1485)

1570

Negative
Test
Type II Error
(15% of 100)
15
True Negative
(85% of 9900)
8415

(15+8415)

Total
Tests 100 9,900 10,000

What is the probability that a
participant actually has the

condition
(so we need to take action) ?

true positives/total positives
and

false negatives/total negatives

Probability that a Positive Result is correct
(true positive)

• Overall, the diagnostic test gives
• positive results to 1570 people (85+1485)

• 85 people who actually have the condition (correct) and to
• 1485 people who do not have the condition (incorrect)

• 85 of these are true positives,

• So the probability of a positive result actually being
correct is 85/1570

Step 5. Calculate Probability that
a Positive Result is correct

True Positives/Total Positives

85 / 1570 = .054 or 5.4%

Has Condition Does not have
Condition

Total
Positive
Test
True Positive
(85% of 100)
85
Type I Error
(15% of 9900)

1485
1570

Probability that a Negative result is incorrect
(false negative)

• Suppose you are a doctor seeing a patient with symptoms of
Condition D.

• The diagnostic test comes back
“not Condition D”.

• Based on the matrix values, what is the chance that the patient
really has Condition D ?

Probability that a Negative result is incorrect
(false negative)

• Overall, the diagnostic test gives
– negative results to 8430 (15+8415) people

• 15 people who do have the condition (incorrect)
• 8415 people who actually do not have the condition (correct)

• 15 of these are false negatives,

• So the probability of a negative result actually being
incorrect is 15/8430

Step 6. Calculate the Probability that a
Negative result is incorrect

False Negatives/ Total Negatives

15 / 8430 � .0018 or .18%

Has Condition Does not have
Condition
Total
Negative
Test
Type II Error
(15% of 100)
15
True Negative

(85% of 9900)
8415 8430

What’s the big deal ?

• False positives (Type I error) lead to un-
needed efforts to “cure” the problem

• False negatives (Type II error) give a false
sense of security and no effort to “cure”
the problem

Possible Combinations and Value
when throwing Two Dice

Die 1

D
ie
2

Score Die 1

Sc
o
re
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 score of both dice

LSP 1

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

,
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 =

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 =

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 =

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 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

26 10

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,

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

% 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 .1

7 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,

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

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 =

/100 = .99
• The number of trials = 15

• P(at least one 100-year flood in 15 years) =
1 – (0.99)15 = 1 – 0.860058 = 0.139

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

Probability of an
Event Happening Exactly Once

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

( P(yes) * [ P(no)] number of trials-1) * number
of trials

Probability of an
Event Happening Exactly Once
• What is the probability of an outcome
happening exactly once in a number of
trials ?

• For example:
– If the probability of an outcome is 0.3 in one trial, what

is the probability of an outcome occurring exactly
once in four trials ?

P(exactly one yes in 4 trials) =
[P(yes) * P(no) * P(no) * P(no) ] * 4

100

Probability of an
Event Happening Exactly Once

If the probability of an outcome is 0.3 in one trial, what is
the probability of an outcome occurring exactly once in four
trials ?

Here are the four possible combinations:
[P(yes) * P(no) * P(no) * P(no)]
[P(no) * P(yes) * P(no) * P(no)]
[P(no) * P(no) * P(yes) * P(no)]
[P(no) * P(no) * P(no) * P(yes)]

101

Probability of an
Event Happening Exactly Once

If the probability of an outcome is 0.3 in one trial, what is
the probability of an outcome occurring exactly once in four
trials ?
Here are the four possible combinations:

A. [P(.3) * P(.7) * P(.7) * P(.7)] = 0.1029
B. [P(.7) * P(.3) * P(.7) * P(.7)] = 0.1029
C. [P(.7) * P(.7) * P(.3) * P(.7)] = 0.1029
D. [P(.7) * P(.7) * P(.7) * P(.3)] = 0.1029

This is an Either/Or case…A or B or C or D
102

Probability of an
Event Happening Exactly Once
If the probability of an outcome is 0.3 in one trial, what is
the probability of an outcome occurring exactly once in four
trials ?

Here are the four possible outcomes:
A. [P(.3) * P(.7) * P(.7) * P(.7)]
B. [P(.7) * P(.3) * P(.7) * P(.7)]
C. [P(.7) * P(.7) * P(.3) * P(.7)]
D. [P(.7) * P(.7) * P(.7) * P(.3)]

103

Probability of an
Event Happening Exactly Once

This is an “Either/Or” case…A or B or C or D could
occur
So the probability of A or B or C or D =

P(A) + P(B) + P(C) + P(D)

Here are the four possible outcomes with probabilities:
A. [P(.3) * P(.7) * P(.7) * P(.7)] = 0.1029
B. [P(.7) * P(.3) * P(.7) * P(.7)] = 0.1029
C. [P(.7) * P(.7) * P(.3) * P(.7)] = 0.1029
D. [P(.7) * P(.7) * P(.7) * P(.3)] = 0.1029

P(A or B or C or D) = .4

116

104

Probability of an
Event Happening Exactly Once

If the probability of an outcome is 0.3 in one trial, what is
the probability of an outcome occurring exactly once in four
trials ?

In this example, the Number of Trials = 4

In this example, probability of exactly one is
[0.3 * 0.7 * 0.7 * 0.7] * 4 = 0.1029 * 4

= 41.16 % as a percent with two decimal places

105

Probability of an
Event Happening Exactly Once

106

• What is the probability that a region will experience
exactly one 100-year flood during the next 25 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 = 25

( P(yes) * [ P(no)] number of trials-1) * number of trials

• P(exactly one 100-year flood in 25 years) =
.01 * (0.99)25-1 = .01 * 0.785678 * 25

= 0.19642
Percentage to three decimal places =19.642%

Calculation of Probability
Gambler’s Fallacy

• You are playing craps in Vegas.
– You have had a string of losses.
– You figure since your luck has been so bad, it has to

balance out and turn good
Bad assumption!

• This is an example of Gambler’s Fallacy
– specifically the “just world” hypothesis

– Each event is independent of another and has
nothing to do with the previous run. Especially in the
short run

107

Gambler’s Fallacy – example

• The probability of winning at any one spin at
roulette is 0.473

• What is the probability of winning at roulette six
spins in a row ?

0.473 * 0.473 * 0.473 * 0.473 * 0.473 * 0.473

(0.473)6 = 0.

111

9868

108

What are the probabilities for
the next spin ?

• after six winning spins

– probability of winning on the next spin is
0.473

– probability of losing on the next spin is
1.0 – 0.473 = 0.527

• the prior winning spins have no impact on the
probabilities of this single spin

109

Expected Value

• Uses
– probabilities of possible outcomes
– and the dollar values of the possible

outcomes

to estimate values that can be compared and
used to make decisions about whether a course
of action is advantageous

111

Expected Value

• A practical (rational)way to determine the dollar
value of outcomes and compare them

Calculated by

Dollar value of the outcome of an event
multiplied by

the probability that the outcome will occur

112

Expected value
Example #1

Example:
I receive a raffle ticket for my birthday. The raffle prize is
$1000. The probability that I will win is 1/10000.

The expected value to me is

+$1000 *1/10000 = $1000/10000 = +$.10

113

Expected value
Example #1 continued

Fred offers to buy my raffle ticket for $1.

The expected value of that transaction to me is

+$1 x 1.00 = +$1.00

114

Should I sell the raffle ticket to Fred ?

compare expected value of keeping ticket with expected
value of selling ticket

Keep ticket
(expected value of raffle ticket to me � +$.10)

Sell ticket
(expected value of what Fred will pay me for it � +$1.00)

Choose the outcome with the higher expected value

115

Calculating Expected Value

• What if there are multiple related events?
– What is the expected value from the set of

events?

• Expected value =
event 1 value x event 1 probability

+ event 2 value x event 2 probability
+ …

116

A tip

• Make sure you keep signs for each dollar
value straight

• Money coming to you (or your
organization) = +

• Money going from you (or your
organization) = –

117

Expected Value

• Two points of view:
– Value to the “buyer”

• money has come out of your pocket (-)

– Value to the “seller”
• money is going into your pocket (+)

118

Expected Value
Buyer – Example #1

Raffle ticket example:

What is the expected value of a raffle ticket ?

A. The ticket costs $5.00
B. The raffle prize is $10,000. The probability that a given ticket will win is
1/10000.

Expected value
A. – $5.00 x 1.00 = – $5.00
B. + $10,000 x 1/10000 = +$1.00
Expected value = ( – $5.00 )

+( +1.00 )
– $4.00

Money out of your pocket
To buy the ticket, you must pay,

so probability is 100% or 1.0

119

On the average, you would lose $4.00
per ticket purchased.
You should not buy this ticket

Expected Value
Buyer – Example #2

• Suppose that a $1 lottery ticket has the
following probabilities:
– 1 in 5 win a free $1 ticket;
– 1 in 100 win $5;
– 1 in 100,000 to win $1000;
– 1 in 10 million to win $1 million.

– What is the expected value of this lottery
ticket?

– Should you buy this ticket ?
120

Expected Value
Buyer – Example #2

• Ticket purchase: value -$1, prob 1.0
• Win free ticket: value $1, prob 1/5
• Win $5: value $5, prob 1/100
• Win $1000: prob 1/100,000
• Win $1million: prob 1/10,000,000

-$1 x 1= -1;
$+1 x 1/5 = +$0.20;
$+5 x 1/100 = +$0.05;
$+1000 x 1/100,000 = +$0.01;
$+1,000,000 x 1/10,000,000 = +$0.10

121

Expected Value
Buyer – Example #2

• Now sum all the products:
-$1 x 1 = -1.00
+$1 x 1/5 = +0.20
+$5 x 1/100 = +0.05
+$1000 x 1/100,000 = +0.01
+$1,000,000 x 1/10,000,000 = +0.10

total -$0.64

Thus, averaged over many tickets, you should expect to
lose $0.64 for each lottery ticket that you buy.

122

Expected Value
Buyer – Example #2
Thus, averaged over many tickets, you should expect to
lose $0.64 for each lottery ticket that you buy.

Should you buy this ticket ?
• The expected value for this lottery ticket is less than zero

(negative value -$.64) so you should not buy this ticket.

• If the expected value were greater than zero (positive),
you should buy the ticket.

123

Expected Value – Seller Example

• Suppose an insurance company (Company A) sells policies for
$1000 each.

• The company knows that 10% of its policy holders will submit a
successful claim that averages $2500 each.

• The company knows that an additional 10% of its policy holders will
submit a successful claim that averages $5000 each.

• How much can the company expect to make per customer?

• If the company sold 100 policies, how much would it expect to
gain/lose ?

124

Expected Value – Seller Example

• Company A takes in (+) $1000 100% of the time (when a
policy is sold)

• Company A pays out (-) $2500 10% of the time

• Company A pays out (-) $5000 10% of the time

125

Expected Value – Seller Example

+ $1000 x 1.0 = + $1000
– $2500 x 0.1 = – $ 250
– $5000 x 0.1 = – $ 500

total = +$250

• Company A would expect to make $250 from each
customer

• If Company A sells 100 policies, it expects to make
$25,000 126

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