Transcript Probability

Chapter 4, Part 1
Basic ideas of Probability
Relative Frequency, Classical Probability
Compound Events, The Addition Rule
Disjoint Events
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Idea of Probability
• Probability is the science of “random”
phenomena or “chance” behavior.
• Many phenomena are unpredictable
when observed only once, but follow a
general pattern if observed many times.
• Examples of “random” phenomena:
– Coin flips, rolling dice, drawing cards
– Drawing names/numbers out of a hat
– ** Choosing a sample from population **
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Modeling Random Behavior
• When we observe “random” behavior, we
do not try to predict the results of a single
observation. Instead, we…
– Consider every possible result that could
happen on a given observation. These are
called outcomes for the random
phenomenon.
– Measure the “chance” that a given outcome
will occur on a particular observation. This is
called a probability.
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Basic Terminology
• An event is any set of outcomes or results
of a given random phenomenon.
• An outcome (or simple event) is a single
result that cannot be broken down into
simpler components.
• The Sample Space for a random
phenomenon is the set of all possible
outcomes.
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Basic Terminology, Examples
• I randomly select one student from my class list.
• Some (non-simple) events:
–
–
–
–
The student is in Row 1
The student is absent
The student is female
The student’s Exam 1 score was above 90%.
• Each individual student on the class list is an
outcome (simple event).
• The Sample Space is the set of all students on
my class list.
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Probability
• When we try to model/describe a random
phenomenon, each event is assigned a
number, called the probability of the event.
• Probability measures how likely it is that
an event will occur. Events with higher
probability are more likely to happen (or
tend to happen more frequently)
• P(A) denotes the probability of event A.
We always require that 0 ≤ P(A) ≤ 1.
Possible Values
for Probabilities:
0 ≤ P(A) ≤ 1
Three Views of Probability
• Relative Frequency (Empirical Probability):
– Actual Data is used to estimate the probability of
various events.
• Classical Approach (Theoretical Probability):
– Assign probabilities in a way that satisfies a set of
formal mathematical rules.
• Subjective Probability (“Expert Opinion”):
– Use prior knowledge from a similar situation in order
to estimate probabilities.
Relative Frequency
• Estimate probability from actual data.
• Take many observations of a random
phenomenon, and count how many times
a particular event occurs.
• The relative frequency of the event is:
(# of occurrences) / (# of observations).
• In other words, for what proportion of
observations did the event occur?
• It may be helpful for you to think of this
as a percentage.
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Relative Frequency: Examples
• Earlier in the class, you drew a slip of
paper from the hat. This gives us about 30
observations of a random phenomenon.
• Using our actual data, we now compute:
• Relative frequency of “Blue” =
• Relative frequency of “Pink” =
Classical Probability
• Our random phenomenon has Sample Space
with finitely many outcomes, say n of them.
• Assume that each outcome is equally likely to
occur on any given observation.
• The (classical) probability of the event is:
(# of outcomes in the event)/(total # of outcomes)
• NOTE: This is actually just a special case of a
more general approach (theoretical probability).
Classical Probability, Examples
• I randomly select one student from class
(among those currently present).
• Assume that each student (outcome) is
equally likely.
• Compute the (classical) probability of:
– The student is in Row 1.
– The student is in the back row.
– The student is texting on his/her cell phone.
Why Classical Probability?
• Suppose our random phenomenon is “Choose a
sample of N individuals from a large population.”
• Each outcome is a group of N individuals.
• Note that N is NOT the total number of
outcomes (that number is MUCH BIGGER)
• If we assume that “all outcomes are equally
likely,” then we are talking about…?
The Law of Large Numbers
• Relative Frequency: Estimate probability
using actual observational data.
• Classical Probability: Compute using
knowledge of the Sample Space.
• Question: What if our knowledge of the
Sample Space is incomplete?
– Example: We know the hat has only pink and
blue slips, but we don’t know how many of
each kind.
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The Law of Large Numbers
• Under a certain condition (independent
observations, discussed next time):
As we increase the number of observations, the
Relative Frequency of an event tends to be closer
to the (theoretical) Probability of that event.
• Relative Frequency estimates Probability. With
more observations, you are more likely to get a
better estimate.
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Compound Events
• Let A, B be two events. For example, if I choose
a student from those in class:
– A = “The student I choose is male.”
– B = “The student I choose is in the back row.”
• Each event is actually a set of outcomes, but
you can think of each event as some kind of
condition/requirement.
• The event “A or B” is the set of outcomes that
meet at least one (but possibly both) of the given
requirements.
Compound Events
• The event “A and B” is the set of outcomes
that meet both of the given requirements.
• In the previous example:
– “A or B”: The chosen student is male, in the
back row, or both (at least one condition is
met).
– “A and B”: The chosen student is male AND in
the back row (both conditions are met).
The Addition Rule
• If A and B are any events, then we have:
P(A or B) = P(A) + P(B) – P(A and B)
• Here’s a version that’s useful when “all
outcomes are equally likely”: Let #(A) be
the number of outcomes in event A. Then
#(A or B) = #(A) + #(B) - #(A and B)
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Disjoint Events
• If events A and B have no outcomes in
common (they cannot occur at the same
time), then we say that they are disjoint.
– Example: “Student is in Row 1” and “Student
is in Row 3”
• In this case P(A and B) = 0. So the
Addition Rule becomes:
P(A or B) = P(A) + P(B)
Complementary Events
• Given an event A, the complement of A is
the set of outcomes that are not in A.
– Notation:
A
is the complement of A.
• Example: Choose a student from class:
– A = “Student is in Row 1”
– B = “Student is Female”
• Note that an event and its complement will
always be disjoint.
Formulas for Complementary
Events
P( A)  P( A)  1
P( A)  1  P( A)
P( A)  1  P( A)