Transcript probability model

We should appreciate, cherish and cultivate blessings.
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Chapters 10, 12: Probability
Sample
space/event
Probability models
Basic probability rules
Random variables
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Random Phenomenon & Probability
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A random phenomenon is one in which the
outcome is unpredictable. The outcome is
unknown until we observe it
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The probability of any outcome of a random
phenomenon is the proportion of times the
outcome would occur in a very long series of
repetitions
Example 1: Traffic Jam on the Golden
Bridge
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< = 20 mph  traffic jam
How often will a driver encounter traffic jam
on the golden bridge during 7-9 am
weekdays? __ out of 100 times.
Example 2: Coin Flipping
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What is the probability that a flipped coin
shows heads up?
Two Ways to Determine Probability
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Making an assumption about the physical
world and use it to find proportions
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Observe outcomes and find its proportions in
a very long series of repetitions
Probability Models
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Sample space: the collection of all possible
outcomes of a random phenomenon
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An event is an outcome or a set of outcomes of a
random phenomenon; i.e. a subset of the sample
space
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A probability model consists of two parts: a sample
space S and a way of assigning probabilities to
events; i.e. a model for distribution
** what are the sample space/event of the examples?
Rules for a Probability Model
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The probability P(A) of any event A is
between 0 and 1
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The probability P(S) of sample space is 1
Building a Probability Model
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(Theoretical way) Making an assumption about the
physical world and use it to build the model
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(Data way) Measuring a representative sample and
observing proportion of the sample that fall into
various outcomes
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Do example 2 and then example 3
Example 3: Employment
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What is the probability that a CSUEB
graduate gets hired within 3 months after
graduation? Between 3 to 6 months? More
than 6 months?
** Data: In a representative sample of 200
graduates, 150, 30, 20 of them got hired
within 3, 3-6, and > 6 months, respectively.
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Probability of an Event
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The sum of the probabilities of outcomes in
the event
** Revisit the examples 2, 3 and find the events
& their probabilities
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Mutually Exclusive Events
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Two events are mutually exclusive if they do
not contain any of the same outcomes. They
are also called disjoint.
Basic Probability Rules (p. 269)
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Rule 3: If two events A and B are disjoint, then
P(A or B) = P(A)+P(B)
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Rule 4: For any event A,
P(A does not occur) = 1- P(A)
** The event (A or B) occurs if either A or B or both occur.
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Probability Models
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A probability model with a finite sample space is
called discrete
Example: problem 10.11 (p. 272)
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A continuous probability model assigns probabilities
as areas under a density curve. The area under the
curve and above any range of values is the
probability of an outcome in that range
Example: randomly pick a number between 0 and 1
Random Variables
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A random variable is a variable whose value is a
numerical outcome of a random phenomenon
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The distribution of a random variable X tells us what
values X can take and how to assign probabilities to
those values
Example:
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1. # of dots (problem 10.11) and
2. height of a young lady (example 10.9)
Independent Events
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Two events are independent if the probability
that one event occurs stays the same, no
matter whether or not the other event occurs.
Conditional Probability
The conditional probability of the event B
given the event A, denoted P(B|A), is the
long-run relative frequency with which event
B occurs when circumstances are such that
event A also occurs.
** event A = age 21+; event B = female
** Ask students for age and find P(B|A)
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Basic Rules for Finding Probabilities
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P(A or B) = P(A) + P(B) - P(A and B)
k disjoint events:
P(A1 or A2 or … or Ak) = P(A1) + P(A2) + … + P(Ak)
P(A and B) = P(A)P(B|A)
k independent events:
P(A1 and A2 and … and Ak) = P(A1)P(A2)…P(Ak)
Steps for Finding Probabilities
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Identify random phenomenon
Identify the sample space
Build the probability model as much as you can
Specify the event for which the probability is
wanted
Use the probability model from step 3 and the
probability rules to find the probability of interest
Tree diagrams
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For a sequence of events, when conditional
probabilities for events based on previous
events are known
Example:
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People are classified into 8 types. For
instance, Type 1 is “Rationalist” and applies
to 15% of men and 8% of women. Type 2 is
“Teacher” and applies to 12% of men and
14% of women. Each person fits one and
only one type.
•What is the probability that a randomly selected
male is “Rationalist”? “Teacher”? Both?
•What is the probability that a randomly selected
female is not a “Teacher”?
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Suppose college roommates have a particularly hard time getting
along with each other if they are both “Rationalists.”
A college randomly assigns roommates of the same sex.
What proportion of male roommate pairs will have this problem?
What proportion of female roommate pairs will have this problem?
Assuming that half of college roommate pairs are male and half are female.
What proportion of all roommate pairs will have this problem?
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