Data Analysis (Quantitative Methods)

1
Data Analysis(Quantitative Methods)

• Lecture 2
• Probability

2
Probability

• An experiment is an act or process of observation
  that leads to a single outcome that cannot be
  predicted with certainty.
• A sample point is the most basic outcome of an
  experiment. Ob1, Ob2, , Obn.

3
Sample Space Venn Diagram

• The sample space of an experiment is the
  collection of all its sample points.
• S Ob1, Ob2, , Obn
• Venn diagrams.
• Graphical representations.

Ob1 Ob2 Obn
S
4
Examples

• Experiment Observe the up face on a coin
• Sample Space 1. Observe a head H
• 2. Observe a tail
  T
• S H, T

H T
S
5
Examples

• Experiment Observe the up face on a die.
• Sample Space 1. Observe a 1.
• 2. Observe a 2.
• 3. Observe a 3.
• 4. Observe a 4.
• 5. Observe a 5.
• 6. Observe a 6.
• S 1,2,3,4,5,6

1 2 3 4 5 6
s
6
Examples

• Experiment Observe the up face on two coins
• Sample Space 1. Observe HH
• 2. Observe HT
• 3. Observe TH
• 4. Observe TT
• S HH,HT,TH,TT

HH HT TH TT
S
7
Probability Rules for Sample Points

• All sample point probabilities must lie between
  0 and 1.
• The probabilities of all the sample points within
  a sample space must sum to 1.

8
Probability

• An Event is a specific collection of Sample
  points.
• Example Consider the experiment of tossing two
  balanced coins.
• Events
• A Observe exactly one head
• B Observe at least one head

9
Probability

• Sample point Probability
• HH 1/4
• HT 1/4
• TH 1/4
• TT 1/4
• P(A)P(HT)P(TH)1/2
• P(B)P(HH)P(TH)P(HT)3/4

10
Probability of an Event

• The probability of an event A is calculated by
  summing the probabilities of the sample points in
  the sample space for A.

11
Steps for Calculating Probabilities of Events

• Define the experiment.
• List the sample points. Ob1,Ob2,,Obn
• Assign probabilities to the sample points.
• P(Ob1), , P(Obn).
• Determine the collection of sample points
  contained in the event of interest.
• Sum the sample point probabilities to get the
  event probability.

12
Steps for Calculating Probabilities of Events

• Example
• Experiment Observe the up face on a die.
• Find the probability of event of sum of two
  throws is equal to 6.
• Solution Sample points 36 (6 by 6).
• A sum of two throws is equal to 6.
• (1,5),(2,4),(3,3),(4,2),(5,1)
• Pr(A)5/36

13
Unions and intersections

• The Union of two events A and B is the event that
  occurs if either A or B or both occur on a single
  performance of the experiment, denoted as the
  symbol

14
Unions and intersections

• The intersection of two events A and B is the
  event that occurs if both A and B occur on a
  single performance of the experiment, denoted as
  the symbol
• P(A ? B)
• Events A and B are mutually exclusive if
• A ? B contains no sample points, i.e. if A and B
  have no sample points in comon.

15
Unions and intersections
A
A
B
16
Unions and intersections

• Example 1.
• Consider the die-toss experiment. Define the
  following events
• A Toss an even number
• B Toss a number less than or equal to 3
• Find

17
Unions and intersections
18
Complementary Events

• The complement of an event A is the event that A
  does not occur -- that is, the event consisting
  of all sample points that are not in event A and
  denoted as symbol Ac
• P(A)P(Ac)1

19
Probability

• Additive Rule of Probability

20
Conditional Probability

• To find the conditional probability that event A
  occurs given that event B occurs, divide the
  probability that both A and B occur by the
  probability that B occurs, that is,

21
Probability

• Tree diagram

HH
H
T
H
HT
TH
H
T
TT
T
22
Independent Events

• Events A and B are independent events if the
  occurrence of B does not alter the probability
  that A has occurred that is, events A and B are
  independent if
• P(AB)P(A)
• When events A and B are independent, it is also
  true that P(BA)P(B)
• Events that are not independent are said to be
  dependent.

23
Probability

• Probability of Intersection of Two independent
  Events
• If events A and B are independent, the
  probability of the intersection of A and B equals
  the product of the probabilities of A and B that
  is P(A ? B)P(A) P(B)
• The converse is also true
• if P(A ? B)P(A) P(B), then A and B are
  independent.

24
Example

• Use tree-diagram to obtain the Sample space of an
  experiment that consists of a fair coin being
  tossed three times. Consider the following
  events
• AAll three results are the same.
• Bexactly one Head occurs.
• Cat least two Heads occur.
• Find P(A),P(B),P(C), P(A)P(B)P(C), P(A? C),
  P(A? B), P(Ac),P(AC)
• Hence, explain if all the events A,B and C are
  not mutually exclusive and independent as well.

25
Solution

• P(A)1/4P(B)3/8P(C)1/2 P(A)P(B)P(C)1,
  P(A? C)1/8, P(A? B)P(A)P(B)-P(A ? B)1/2.
• P(Ac)1-1/43/4,P(AC) P(A? C)/P(C)1/4
• P(A)P(B)3/32 ? P(A ? B) (P(A ? B) 0)
• P(A)P(C)1/8 P(A ? C)
• P(B)P(C)3/16 ? P(B? C) (P(B ? C) 1/8)

26
Exercise

• Calculate the mode, mean, and median of the
  following data
• (1) 12, 13, 15, 18, 12, 56, 13, 17, 19, 20, 35,
  36
• (2) 35, 23, 18, 26, 35, 23, 39, 45, 47, 37, 23,
  35, 19

27
Answer

• (1)
• Mode 12,13
• Mean 22.17
• Median 17.5
• (2)
• Mode 23, 35
• Mean 31.15
• Median 35

28
Excercise

• Calculate the range, variance and standard
  deviation of the following data
• (1) 2, 3, 1, 6, 8, 5, 9, 4, 5
• (2) 2, 0, 8, 4, 7, 5, 3, 2, 100

29
Answer

• (1)
• range 8
• sample variance 6.94
• sample standard deviation 2.64
• (2)
• range 98
• sample variance 1159.32
• sample standard deviation 34.05

30
Excercise

• Two fair coins are tossed and the following
  events are defined
• AObserved at least one head
• BObserved exactly one head
• CObserved exactly one tail
• DObserved at most one head
• Find P(A), P(B ? D), P(AD)

31
Answer

• Pr(A)3/4
• Pr(B ? D)1/2
• Pr(AD)2/3

32
Exercise

• Use tree-diagram to obtain the Sample space of an
  experiment that consists of a fair coin being
  tossed four times. Consider the following
  events
• AAll four results are the same.
• Bexactly one Head occurs.
• Cat least two Heads occur.
• Find P(A),P(B),P(C), P(A)P(B)P(C), P(A? C),
  P(A? B)
• Hence, explain why all the events A,B and C are
  not mutually exclusive.

33
Answer

• Pr(A)1/8, Pr(B)1/4,Pr(C)11/16
• Pr(A)Pr(B)Pr(C)17/16
• Pr(A?C)1/16
• Pr(A?C)3/4
• Since Pr(A?C) is not equal to 0, so A,B,C are
  not mutually exclusive.
• But Pr(A ?B)Pr(B ?C)0, A,B and B,C are mutually
  exclusive respectively.

34
Exercise

• Let P(A)0.7, P(B)0.5 and P(A? B) 0.8.
• Find (1) P(A? B) (2) P(BA)

(3) Is event A independent of event B?
35
Answer

• Pr(A ?B)Pr(A)Pr(B)-Pr(A?B)0.4
• Pr(BA) Pr(A ?B)/Pr(A)4/7
• Since Pr(BA) is not equal to Pr(B), so event B
  is not independent of A.

36
References

• Statistics, 8th Edition
• MaClave and Sincich
• Prentice Hall, 2000.