Chapter 4 PROBABILITY

1
Chapter 4PROBABILITY

• Mrs. Mandy Wimpey
• Palmetto High School

2
Real-Life Application

• A woman you know has taken a pregnancy test.
• What is the chance the test provides a false
  positive result she is not pregnant, but the
  test indicates pregnancy?
• What is the chance the test provides a false
  negative resultshe is pregnant, but the test
  indicates no pregnancy?

3
The Fundamentals

• EVENT a collection of results or outcomes of a
  procedure the thing that happens
• SIMPLE EVENT an outcome that cannot be broken
  into simpler part
• COMPOUND EVENT an outcome that can be broken
  into other events
• SAMPLE SPACE the collection of all possible
  events

4
Notation

• P means probability
• P(A) means the probability of event A occurring

5
The Formula

• Round probabilities to 3 significant digits.

6
Example

• A couple wants 3 children.
• Find the sample space.
• Find P(all girls)
• Find P(1 girl)
• Find P(at least 2 girls)

7
Example
The possibilities for having the BOY first.

• B
• B
• G
• B
• B
• G
• G

8
Example
The possibilities for having the GIRL first.

• B
• B
• G
• G
• B
• G
• G

9
Example

• So, the sample space is as follows
• S BBB, BGB, BBG, BGG, GGG, GBG, GBB, GGB
• There are 8 total possible combinations of three
  children.

10
Example
S BBB, BGB, BBG, BGG, GGG, GBG, GBB, GGB

• Given the sample space, we can find the
  probabilities now.
• P(all girls) of times GGG occurs
• total of possible outcomes
• 1
• 8

S BBB, BGB, BBG, BGG, GGG, GBG, GBB, GGB
11
Example
S BBB, BGB, BBG, BGG, GGG, GBG, GBB, GGB

• P(1 girl) of times one girl occurs
• total of possible outcomes
• 3
• 8

12
Example
S BBB, BGB, BBG, BGG, GGG, GBG, GBB, GGB

• P(at least 2 girls) of times at least two
  girls occurs
• total of possible outcomes
• 4
• 8

13
The Law of Large Numbers

• When an experiment is repeated many, many times
• OBSERVED THEORETICAL

14
Special Probabilities

• IMPOSSIBLE EVENTS
• Those events that will not occur
• Probability 0
• SURE EVENTS
• Those events that are certain to occur
• Probability 1.0 or 100

15
The probability of an event will always be
between 0 and 1, inclusively.
16
Complement

• If A is the event, then not A is its
  complement.
• Example There is a 70 chance of rain today.
  The complement is not rain.
• So, there is a 100 - 70 30 chance of it not
  raining today.
• Complements are found by subtracting from 100,
  or 1.

17
Complement

• In reality, there are more boys born than girls.
  Statistically, there will be 105 boys born out of
  every 205 births.
• What is the probability for a girl being born?

18
Complement

• The probability of a girl the probability of
  NOT having a boy.
• P(boy) 105/205 0.512
• So, P(not boy) 1-0.512 0.488

19
The Addition Rule for Probability

• The Addition Rule is used for situations
  involving the word OR.
• You do not want both events to occur, but you
  want one of them to occur.

20
The Addition Rule
You have to be careful not to double count
anything!
21
The Addition Rule
The results from the sinking of the Titanic.
22

• Example 1
• P(man or survived)
• P(man) P(survived) P(both)
• ( men / total) ( survived / total) P(men
  who survived)
• 1692/2223 706/2223 332/2223
• 0.929
• Example 2
• P(woman or child)
• P(woman) P(child)
• ( women / total) ( children / total)
• 422/2223 109/2223 0/2223 0.239

23
The Multiplication Rule for Probability

• The Multiplication Rule is used in situations
  involving the word AND.
• You want all events to occur, not just one or the
  other.

24
The Multiplication Rule
Event A occurs, then event B occurs
25
With Replacement

• The first item is replaced into the sample space
  before the next item is drawn
• The events are independent of each other the
  occurrence of one does not have any affect on the
  probability of the occurrence of the next one

26
Without Replacement

• The first item is NOT replaced into the sample
  space before the next item is drawn.
• Therefore, the next item is drawn with the first
  item being absent from the sample space.
• The events are NOT independent.

27

• Example 1 (with replacement)
• A bag of marbles has 3 red marbles, 2 blue
  marbles, 4 yellow marbles, and 1 green marble.
  One marble is drawn and replaced before drawing
  again.
• Find the probability of drawing a red marble the
  first time and a blue one the second time.
• P(red and blue)
• P(red) x P(blue)
• ( red / total) x ( blue / total)
• (3/10) x (2/10)
• 0.0600

28

• Example 2 (w/o replacement)
• A bag of marbles has 3 red marbles, 2 blue
  marbles, 4 yellow marbles, and 1 green marble.
  One marble is drawn but is NOT replaced before
  drawing again.
• Find the probability of drawing a red marble the
  first time and a blue one the second time.
• P(red and blue)
• P(red) x P(blue)
• ( red / total) x ( blue / total)
• (3/10) x (2/9)
• 0.0667

29

• Example 3
• Seventy-nine percent of all college freshmen live
  on campus. If you were to randomly select 3
  college freshmen, what is the probability that
  all 3 of them live off campus?
• We are given the that live on campus. So,
• P(live off) 1 P(live on)
• 1- 0.79 0.21 live off campus
• Since we want all 3 we select at random to live
  off campus, we have
• P(1st lives off AND 2nd lives off AND 3rd lives
  off)
• P(1st) x P(2nd ) x P(3rd)
• 0.21 x 0.21 x 0.21
• 0.00926

30

• Example 4
• You bought a new computer and want to set a
  password for log-in. Your password needs to be 4
  characters consisting of letters and digits.
  Repetition is allowed.
• What is the probability your password is R2D2?
• Since there are 4 characters, you must find the
  probability of each character and multiply them
  together.
• P(R) 1/26
• P(2) 1/10
• P(D) 1/26
• So, the probability P(R) x P(2) x P(D) x P(2)
• (1/26) x (1/10) x (1/26) x (1/10)
• 1.479 x 10-5
• 0.00001479

31

• Example 5 (no repetition allowed)
• You bought a new computer and want to set a
  password for log-in. Your password needs to be 4
  characters consisting of letters and digits.
  Repetition is NOT ALLOWED.
• What is the probability your password is R2D2?
• Since there are 4 characters, you must find the
  probability of each character and multiply them
  together.
• So, the probability P(R) x P(2) x P(D) x P(2)
• (1/26) x (1/10) x (1/25) x (1/9)
• 1.709 x 10-5
• 0.00001709

32
Conditional Probability
intersection
given
33
Results from a study conducted of Whether a skin
cream improved eczema
This table is called a contingency table. This
one uses counts.
34

• P(improved used cream)
• P(improved and used cream)
• P(used cream)
• 800/2000
• 1200/2000
• 800/1200
• 0.667
• So, out of the group that used the cream, 66.7
  saw an improvement in their skin.

35

• P(used cream improved skin)
• P(used cream and improved skin)
• P(improved skin)
• 800/2000
• 1400/2000
• 800/1400
• 0.571
• So, out of the group that saw an improvement in
  their skin, 57.1 of them had used the cream.

36
Diagnostic Testing

• Diagnostic Testing is used to diagnose diseases
  and conditions.
• Pregnancy
• Blood tests
• Drug tests
• HIV testing
• Cancer tests
• Etc.

37
Terms in Diagnostic Testing

• The condition is present is indicated by an S
  or sometimes a D.
• The condition is not present is indicated by an
  S or sometimes a D .
• Positive the test indicates the condition is
  present
• Negative the test indicates the condition is
  not present (or, is absent)
• False Positive the test indicates the condition
  is present when it is absent
• False Negative the test indicates the condition
  is absent when it is present

c
c
38
Terms in Diagnostic Testing

• Sensitivity The probability that the test
  detected the present condition
• P( S)
• Specificity The probability that the test is
  correct in providing a negative result
• P(- S )
• Prevalence Rate The probability that the
  condition is present in the given population
• P(S)

c
39
Example for Drug Diagnostic Testing and
Conditional Probability

• Air traffic controllers are given random urine
  tests for drug abuse. The numbers for the test
  are as follows
• Sensitivity 0.958
• Specificity 0.927
• Prevalence Rate 0.007

40
Example for Drug Diagnostic Testing and
Conditional Probability

• Construct a Tree Diagram to help you construct
  your contingency table.
• DRUG USE? TEST RESULT JOINT PROBABILTIES
• pos
• yes
• neg
• People tested
• pos
• no
• neg

sensitivity
Joint Probabilities are found by
multiplying each branch together
Prevalence rate
Specificity
41
Example for Drug Diagnostic Testing and
Conditional Probability

• Construct a Tree Diagram to help you construct
  your contingency table.
• DRUG USE? TEST RESULT JOINT PROBABILTIES
• pos
• yes
• neg
• People tested
• pos
• no
• neg

0.958
0.007
0.927
42
Example for Drug Diagnostic Testing and
Conditional Probability

• Construct a Tree Diagram to help you construct
  your contingency table.
• DRUG USE? TEST RESULT JOINT PROBABILTIES
• pos
• yes
• neg
• People tested
• pos
• no
• neg

0.958
0.007
1-0.958 0.042
1-0.927 0.073
1-0.007 0.993
0.927
43
Example for Drug Diagnostic Testing and
Conditional Probability

• Construct a Tree Diagram to help you construct
  your contingency table.
• DRUG USE? TEST RESULT JOINT PROBABILTIES
• pos 0.007 x 0.958 0.00671
• yes
• neg 0.007 x 0.042 0.000294
• People tested
• pos 0.993 x 0.073 0.072489
• no
• neg 0.993 x 0.927 0.920511

0.958
0.007
1-0.958 0.042
1-0.927 0.073
1-0.007 0.993

0.927
1.0
44
Contingency Table for Drug Testing of Air Traffic
Controllers
c
45

• P(neg S) 0.000294 / 1.0
• 0.007 / 1.0
• 0.04
• P(pos S ) 0.072489 / 1.0
• 0.993 / 1.0
• 0.073
• P(S pos) 0.00671 / 1.0
• 0.079199 / 1.0
• 0.0847

Meaning Of the group that is doing drugs, there
is a 4 chance they will test negative.
c
Meaning Of the group that is not doing drugs,
there is a 7.3 chance they will test positive.
Meaning Of the group that tests positive, there
is an 8.5 chance they are actually doing drugs.