Introduction To Probability
Introduction To Probability
Introduction To Probability
Chapter 3
Chapter 3 Outline
Introduction W5 L9
Sample Spaces And Probability Concepts
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W5 L9 Learning Outcomes
• Describe basic concepts of probability
• Determine sample spaces for
– Simple event
– Compound event
• Construct tree diagram to show sample space.
• Describe and identify 3 probability approaches
– Classical
– Empirical
– Subjective
• List 4 basic rules of probability
• Construct Venn diagram for intersection and union
events.
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Introduction
• Probability is the chance of an event occurring.
– E.g. a packet of candies…..
• Probability connects Descriptive Statistics and Inferential
Statistics.
• Make decisions about population based on information
from sample.
• Predictions are made based on probability.
• Application: card games, insurance, investment, weather
forecast, etc.
• Data>>Information>>Knowledge
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2 Aspects of Statistics
Application
Development
of those to
& proof of
Statistics solve real
formula,
problems!
theorems, etc
Theoretical Applied
Descriptive Inferential
C.O.A. I.D.c
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Sampling from Population
Sample of Entire population of
observations observations
Random selection
Use
Probability Parameter
Statistic µ=?
X
Make decisions about population based on info from sample
Statistical Inference
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Basic Concepts
• Probability experiment:
– A chance process that leads to well-defined results called
outcomes.
• Outcome:
– The result of a single trial of a probability experiment.
• Sample space:
– A collection of all possible outcomes of a probability experiment.
• Event:
– Any collection of outcomes of a probability experiment.
– i.e. any subset of the sample space is an event.
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Examples
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Example1
• Find the sample space for rolling two dice!
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Tree Diagram
• A diagram to determine all possible outcomes (sample
space) of a probability experiment.
• Example: Tree diagram for the experiment of tossing a
coin twice. (head or tail?)
Outcomes
2nd toss
H HH
1st toss
H
T HT
H TH
T
T TT
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Example 2
• Find the sample space for the gender of three children in
a family. (boy or girl?)
Outcomes
3rd child
BBB
2 child
nd
B
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Solution
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Lets Exercise!
2. You are selected to participate in a contest at a mall.
• You need to choose a pair of shoes and then a pair of
socks.
• There are a total of 4 pairs of shoes and each has a
selection of 3 pairs of socks.
• Draw a tree diagram to show the sample space.
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Solution
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Venn Diagram
• A picture (a closed geometric shape) that shows all the
possible outcomes for an experiment.
• The symbol represents the union of two events and
P(A B) corresponds to A OR B.
• The symbol represents the intersection of two events
and P(A B) corresponds to A AND B.
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Simple Event
• Simple event – an event with only one outcome.
e.g: If a die is rolled and a 6 shows.
• It is also called an elementary event.
• Denoted by E.
• Other examples?
– A coin is tossed and a head shows.
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Compound Event
• Compound Event - an event with more than one
outcome.
e.g : The event of getting an odd number when a die is
rolled.
Since it consists of three outcomes or three simple
events (odd numbers are 1, 3, 5)
• It is also called composite event.
• Denoted by A,B,C,….
• Other examples?
– The event of getting even number when a die is rolled.
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3 Basic Probability Approaches
1. Classical probability
2. Empirical or relative frequency probability
3. Subjective probability
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Classical Probability
• Uses sample spaces to determine the probability an
event will happen.
• Assumes that all outcomes in the sample space are
equally likely to occur; all the events have the same
probability of occurring.
• The probability of any event E is:
• Number of outcomes in E
Total number of outcomes
in the sample space
P( E ) n( E )
n( S )
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Examples
1. When a single die is rolled, each outcome has the same
probability of occurring.
– Since there are six outcomes (1,2,3,4,5,6), each outcome has a
probability of 1/6.
3. There
What is the
are 4 probability
queens, thusof getting
P(queen)a queen?
= 4/52 = 1/13.
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Empirical Probability
• Relies on actual experience to determine the likelihood
of outcomes.
• Is based on observation.
• Given a frequency distribution, the probability of an event
being in a given class is:
• Frequency for the class
• Total frequencies in the distribution
• Or denoted as,
P( E ) n
f
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Example
• Hospital records indicate that maternity patients stayed in the hospital for
the number of days shown in the following distribution:
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Subjective Probability
• Uses a probability value based on an educated guess or
estimate, employing opinions and inexact information.
• This guess is based on the person’s experience and
evaluation of a solution.
• E.g: A physician might say that, on the basis of her
diagnosis, there is a 30% chance the patient will need an
operation.
• What is your chance of scoring A in the mid-sem test?
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Probability Rules
1. The probability of any event E is a number between and
including 0 and 1.
0 P( E ) 1
2. If an event E cannot occur, its probability is 0
(impossible event).
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Intersection Event
• Let A and B be two events defined in a sample space.
• The intersection of events A and B is the event that
occurs when both A and B occur.
• It is denoted by either A B or AB.
• E.g.
• A = event that a family owns a DVD player
B = event that a family owns a digital camera
Family owns both
DVD player
and digital camera)
SQQS1013 W5L9 ZZ 26
Union Event
• Let A and B be two events defined in a sample space.
• The union of events A and B is the event that occurs
when either A or B or both occur.
• It is denoted as A B.
• E.g.
• A = event that a family owns a DVD player
B = event that a family owns a digital camera
All the shaded
areas give the
union of events A
and B.
(A or B or both
A&B)
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Example 8
• A senior citizens centre has 300 members. Of them,
140 are male, 210 take at least one medicine on a
permanent basis and 95 are male and take at least one
medicine on a permanent basis. Draw a Venn diagram
to describe,
a) the intersection of the events “male” and “take at least one
medicine on a permanent basis”.
b) the union of the events “male” and “take at least one medicine
on a permanent basis”.
c) the intersection of the events “female” and “take at least one
medicine on a permanent basis”.
d) the union of the events “female” and “take at least one medicine
on a permanent basis”.
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• Let us define the following events:
• M = a senior citizen is a male
• F = a senior citizen is a female
• A = a senior citizen takes at least one medicine
• B = a senior citizen does not take any medicine
• a) n(M A) = 95
M A
95
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b) n (M A) = 45 + 95 + 115 = 255
M A
45 95 115
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W5 L9: Closure
You should now be able to:
• Describe the basic concepts of probability.
• Determine sample spaces for
– simple event, compound event
• Construct tree diagram to show sample space.
• Describe and identify 3 probability approaches
– classical, empirical, subjective
• List 4 basic rules of probability.
• Construct Venn diagram for intersection and union events.
Next Lesson: W5 L10
• Dependent, Independent Events
• Joint, Marginal, Conditional Probability
• Events & Probability Rules
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