Download Test of a population

Probability
1
4
Probability
.
4.1
Events
Event is an outcome of an experiment or survey.
Such as,
-rolling a die and turning up six dots,
-an individual who votes for the incumbent candidate in an election,
-someone purchasing a specific brand of soft drink.
Elementary Event is an outcome that satisfies only one criterion.
Such as,
-a voter who selected the Democratic candidate,
-a consumer who purchased a Diet Coke soft drink.
Joint Event is an outcome that satisfies two or more criteria.
-a voter who voted for the Democratic candidate for president and the Democrat candidate for U.S. senator,
- a female who purchased a Diet Coke soft drink.
Complement of an event A (denoted A’) All events that are not part of event A .
Such as - females who didn’t purchased a Diet Coke soft drink.
-cards which are not red,
- gender who are not women
4.2
More Definitions
Random variable
A variable whose numerical values represent the events of an experiment.
- scores of students on a standardized exam
- the preferences of consumers for different brands of automobiles.
Probability
A number that represents the chance that a particular event will occur for a random variable. (always between 0
and 1).
-Impossible Event – an event that has no chance of occurring (probability = 0)
- Certain Event – an event that is sure to occur (probability = 1)
Collectively Exhaustive Events
A set of events that includes all the possible events.
4.3
Some Rules of Probability
RULE 1 The probability of an event must be between 0 and 1.
RULE 2 The complement event of A (simply not A), is given the symbol A' and If P(A) represents the probability of
event A occurring, then P(A’)=1 – P(A).
RULE 3 If two events A and B are mutually exclusive (male or female, head or tail etc.), the probability
of both events A and B occurring is 0. This means that the two events cannot occur at the same time.
RULE 4 If two events A and B are mutually exclusive, the probability of either event A or
event B occurring is the sum of their separate probabilities.
P(A OR B) = P(A  B) = P(A) + P(B)
Probability
2
RULE 5 If events in a set are mutually exclusive and collectively exhaustive (either man or women), the
sum of their probabilities must add up to 1.
P(A)+P(A’)=1
RULE 6 If two events A and B are not mutually exclusive, the probability of either event A or event B occurring is
the sum of their separate probabilities minus the probability of their simultaneous occurrence (the joint
probability).
RULE 7 If two events A and B are independent, the probability of both events A and B occurring is equal to the
product of their individual probabilities. Two events are independent if the occurrence of one event in no way
affects the probability of the second event.
RULE 8 If two events A and B are not independent, the probability of both events A and B occurring is the
product of the probability of event A multiplied by the probability of event B occurring, given that event A has
occurred.
WORKED-OUT EXAMPLE 1 A population is described in the following table.
Taking
Stats
Not Taking
Stats
Total
Male
84
145
229
Female
76
134
210
Total
160
279
439
a) Find the probability of selecting a male taking statistics,
b) Find the probability of selecting a female not taking statistics,
c) Find the probability of selecting a person (male or female) taking statistics.
4.4
Sample Space
The Sample Space is the collection of all possible events.
e.g. All 6 faces of a die:
e.g. All 52 cards of a bridge deck:
Experiment
Toss a Coin, Note Face
Toss 2 Coins, Note Faces
Select 1 Card, Note Kind
Select 1 Card, Note Color
Play a Football Game
Inspect a Part, Note Quality
Observe Gender
Sample Space
{Head, Tail}
{HH, HT, TH, TT}
{2♥, 2♠, ..., A♦} (52)
{Red, Black}
{Win, Lose, Tie}
{Defective, Good}
{Male, Female}
Probability
3
Contingency Tables
Decision Trees
Venn Diagrams
WORKED-OUT EXAMPLE 2
Probability
4
WORKED-OUT EXAMPLE 3 Find the probabilities for the 52 cards of a bridge deck.
a)
Probability of selecting a red card and it is ace.
P (Red and Ace) 
numberof cards that are red and ace 2

total numberof cards
52
b) Probability of selecting an ace card.
P(Ace)  P( Ace and Re d )  P( Ace and Black) 
c)
2 2 4
 
52 52 52
Probability of red or ace
P(Red  Ace) = P(Red)+ P(Ace) - P(Red  Ace)
Marginal and Joint Probabilities in a Contingency Table
4.5
General Addition Rule
P(A or B) = P(A) + P(B) - P(A and B)
If A and B are mutually exclusive, then P(A and B) = 0.
WORKED-OUT EXAMPLE 4 What is the probability of red or Ace?
P (Red or Ace) = P (Red) +P (Ace) – P (Red and Ace)
P(Red  Ace) = P(Red)+ P(Ace) - P(Red  Ace)
=
26
52
+
4
52
−
2
52
=
28
52
Probability
5
WORKED-OUT EXAMPLE 5 Using the additive rule, what is the probability of
P (A  D) = P(A or D)
P (B  C) =P (B or C)
4.6
Computing Conditional Probabilities
A conditional probability is the probability of one event, given that another event has occurred:
P(A| B) 
P(Aand B)
P(B)
P(B| A) 
P(Aand B)
P(A)
The conditional probability of A given that B has occurred.
The conditional probability of B given that A has occurred.
Where P(A and B) = joint probability of A and B
P(A) = marginal or simple probability of A
P(B) = marginal or simple probability of B
Presentation of Conditional probability with Venn Diagram
The probability of Ace when black happens is represented as
Event (Ace and Black)
Black ‘Happens’: Eliminates All Other Outcomes.
WORKED-OUT EXAMPLE 6 Of the cars on a used car lot, 70% have air conditioning (AC) and 40% have a CD player
(CD). 20% of the cars have both. What is the probability that a car has a CD player, given that it has AC?
i.e., we want to find P (CD | AC).
Solution:
Using cross-classification table;
or
Of the cars on a used car lot, 70% have air conditioning (AC) and 40% have a CD player (CD).
20% of the cars have both.
Probability
6
P(CD| AC) 
P(CDand AC) 0.2

 0.2857
P(AC)
0.7
Using Venn diagram in conditional probability
AC
CD
0.5
0.2
0.2
0.1
70% have air conditioning (AC) and 40% have a CD player (CD) 20% of the cars have both.
The question is the probability that a car has a CD player, given that it has AC.
The probability to have AC=0.7, and to have both AC and CD player=0.2
P(CD| AC) 
P(CDand AC) 0.2

 0.2857
P(AC)
0.7
Using Decision Trees in conditional probability
WORKED-OUT EXAMPLE 7 For the given events find the conditional probabilities.
a) P(A|D) = (probability of A when D happens)=
b) P(C|B) = (probability of C when B happens)=
Probability
7
4.6 Multiplicative Rule
Used to get compound probabilities for intersection of events.
1. For dependent events
P(A and B) = P(A  B)
= P(A)  P(B|A)
= P(B)  P(A|B)
2. For Independent Events:
P(A and B) = P(A  B)
= P(A)  P(B)
WORKED-OUT EXAMPLE 8 Draw 1 Card. Note Kind & Color.
P(Ace and Black) = P(Ace  Black) = P(Ace)∙P(Black | Ace)
 4  2  2
    
 52  4  52
WORKED-OUT EXAMPLE 9 Using the multiplicative rule, what’s the probability?
P(C  B) =
b. P(B  D) =
c. P(A  B) =
a.
WORKED-OUT EXAMPLE 10 Select 2 pens from 20 pens which have the color of 6 red and 14 blue without
replacing.
a.
b.
c.
Probability of selecting a blue then a red pencil,
Probability of selecting a red then again a red pencil,
Probability of selecting a red then a blue pencil.
Probability
8
Summary of Basic probability
1.
2.
3.
4.
5.
Probability of entire sample: P(entire sample)= 1
For any event A in the sample: 0≤P(A) ≤1
A’ is the complement of A : P(A’)=1-P(A)
A and B are independent events if P(A)= P(A, given B) or
Multiplication rule
General: P (A and B) = P (A) ∙ P (B, given A)
Independent events: P (A and B) = P (A) ∙ P (B)
6.
Conditional probability: P (A, given B)=
7.
8.
Events A and B are mutually exclusive if P(A and B)=0
Addition rule
General: P (A or B) = P (A) +P (B)-P (A and B)
Mutually exclusive events: P (A or B) = P (A) + P(B)
P(A and B)
P(B)
Test Yourself Short Answers
1. If two events are collectively exhaustive, what is the
probability that one or the other occurs?
(a) 0
(b) 0.50
(c) 1.00
(d) Cannot be determined from the information given
2. If two events are collectively exhaustive, what is the
probability that both occur at the same time?
(a) 0
(b) 0.50
(c) 1.00
(d) Cannot be determined from the information given
3. If two events are mutually exclusive, what is the
probability that both occur at the same time?
(a) 0
(b) 0.50
(c) 1.00
(d) Cannot be determined from the information given
4. If the outcome of event A is not affected by event B,
then events A and B are said to be:
(a) mutually exclusive
(b) independent
(c) collectively exhaustive
(d) dependent
Use the following problem description when answering
Questions 5 through 9:
A survey is taken among customers of a fast-food
restaurant to determine preference for hamburger or
chicken. Of 200 respondents selected, 125 were male and
75 were female. 120 preferred hamburger and 80
preferred chicken. Of the males, 85 preferred hamburger.
5. The probability that a randomly selected individual is a
male is:
(a) 125/200
(b) 75/200
(c) 120/200
(d) 200/200
6. The probability that a randomly selected individual
prefers hamburger or chicken is:
(a) 0/200
(b) 125/200
(c) 75/200
(d) 200/200
7. Suppose that two individuals are randomly selected.
The probability that both prefer hamburger is:
(a) (120/200)(120/200)
(b) (120/200)
(c) (120/200)(119/199)
(d) (85/200)
8. The probability that a randomly selected individual
prefers hamburger is:
(a) 0/200
(b) 120/200
(c) 75/200
(d) 200/200
9. The probability that a randomly selected individual
prefers hamburger or is a male is:
(a) 0/200
(b) 125/200
(c) 160/200
(d) 200/200
10. The smallest possible value for a probability is
__________.
11. The largest possible value for a probability is
__________.
12. If two events are _________, they cannot occur at the
same time.
13. If two events are _________, the probability that both
events occur is the product of their individual probabilities.
14. In the _________ probability approach, probabilities
are based on frequencies obtained from surveys.
Probability
9
15. In the __________ probability approach, probabilities
can vary depending on the individual assigning them.
Problems
1. Where people turn to for news is different for various age groups. Suppose that a study conducted on this issue
was based on 200 respondents who were between the ages of 36 and 50, and 200 respondents who were over age
50. Of the 200 respondents who were between the ages of 36 and 50, 82 got their news primarily from newspapers.
Of the 200 respondents who were over age 50, 104 got their news primarily from newspapers. If a respondent is
selected at random, what is the probability that he or she
(a) got news primarily from newspapers?
(b) got news primarily from newspapers and is over 50 years old?
(c) got news primarily from newspapers or is over 50 years old?
(d) Explain the difference in the results in (b) and (c).
(e) Suppose that two respondents are selected. What is the probability that both got news primarily from
newspapers?
2. A survey of 500 men and 500 women designed to study financial tensions between couples asked how likely
each was to hide purchases of clothing from his or her partner. The results were as follows:
Likely to Hide Purchase of
Men
Clothing
Yes
62
No
438
Total
500
If a respondent is chosen at random, what is the probability that
(a) he or she is likely to hide clothing purchases?
(b) the person is a female and is likely to hide clothing purchases?
(c) the person is a female or is likely to hide clothing purchases?
Women
Total
116
384
500
178
822
1,000
3. In 37 of the 58 years from 1950 through 2007, the S&P 500 finished higher after the first five days of trading. In
32 of those 37 years, the S&P 500 finished higher for the year. Is a good first week a good omen for the upcoming
year? The following table gives the first-week and annual performance over this 58-year period: S&P 500’s Annual
Performance
First five days
Higher
Lower
Year
Higher
Lower
32
5
11
10
If a year is selected at random, what is the probability that
(a) the S&P 500 finished higher for the year?
(b) the S&P 500 finished higher after the first five days of trading?
(c) the S&P 500 finished higher after the first five days of trading and the S&P 500 finished higher for the year?
(d) the S&P 500 finished higher after the first five days of trading or the S&P 500 finished higher for the year?
(e) Given that the S&P 500 finished higher after the first five days of trading, what is the probability that it finished
higher for the year?
Additional Questions
4. (3./Basic statistics) Valid probabilities Which of the following numbers cannot be the probability of some
event?
a) 0.71
b) 4.1
c) 1/8
d) -0.5
e) 0.5
f) 0
g) 1
h) 150%
5. ( 6/Basic Statistics) Probability estimate Can you raise eyebrow at a time? Use the students in your statistics
class to estimate the percentage of people who can raise one eyebrow at a time. How can your result be thought of as
an estimate for the probability that a person chosen at random can raise one eyebrow at a time? ( Comment: National
statistics indicate that about 30% of Americans can raise one eyebrow at a time).
Probability
10
Can rise one eyebrow
Boys
Girls
What is the
a) Probability of someone can rise one eyebrow?
b) Probability of a boy can rise eyebrow?
c) Probability of a girl or can rise eye brow?
d) Probability of a girl and cannot rise eye brow?
Cannot rise eyebrow
Probability
11
Probability
12