Transcript Probability

Class 28
Probability
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Causation and Probability
• We are interested in finding the effect of Dr.
Wong’s exploring teaching methodology on the
statistics test score of a sample of TRMC
students.
• Assume we could really isolate all other
lurking variables and examine only the effect
of Dr. Wong’s teaching methodology.
• If there is a gain in the test score, could we
conclude that Dr. Wong’s teaching
methodology works?
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7.1 Random Circumstances
Random circumstance is one in which
the outcome is unpredictable.
Outcome is NOT determined until we observe it.
Alicia has a Bad Day
VS
A 40 year old man dies tomorrow
VS
Get a 2 out of rolling a die
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Randomness and Probability
What is the probability of getting a head when
you toss a coin?
Toss your coins ten times and record heads (H) or
tails (T) on each toss.
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Class Work
What is the probability of getting a head when
you toss a coin?
Toss your coins ten times and record heads (H) or
tails (T) on each toss.
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Theoretical Probability VS
Empirical Probability
• Theoretical Probability
– the number of ways that the desired event can
occur, divided by the total number of possible
outcomes (sample space of known equally likely
outcomes).
• Empirical Probability
– is an "estimate" that the event will happen based on
how often the event occurs after collecting data or
running an experiment (in a large number of trials).
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Randomness and probability
• A random phenomenon: individual outcomes are
uncertain but there is a regular distribution of
outcomes in a large number of repetitions
• Probability:
– The regular distribution of outcomes in terms of a proportion
of times the outcome would occur in a large number of
repetitions.
– The value of probability is between 0 and 1
• Probability describes the long-term regularity of
random phenomenon.
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Assigning Probabilities
• A probability is a value between 0 and 1 and is written
either as a fraction or as a decimal fraction.
• A probability simply is a number between 0 and 1 that
is assigned to a possible outcome of a random
circumstance.
• For the complete set of distinct possible outcomes of a
random circumstance, the total of the assigned
probabilities must equal 1
• . The probability that an event does not occur is 1
minus the probability that the event does occur. If two
events have no outcomes in common, the probability
that one or the other occurs is the sum of their
individual probabilities.
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Classwork: Rolling two dice
1. What are the total possible outcomes?
2. What is the probability of each outcome?
3. What is the probability that the sum of the two dice
is “5”?.
4. What is the probability that the sum of the two dice
is not “5”
5. What is the probability that the sum of the two dice
is “7” or “11”?
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7.2 Interpretations
of Probability
The Relative Frequency
Interpretation of Probability
In situations that we can imagine repeating
many times, we define the probability of a specific
outcome as the proportion of times it would occur
over the long run -- called the relative frequency
of that particular outcome.
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Example 7.1 Probability of Male
versus Female Births
Long-run relative frequency of males
born in the United States is about 0.512.
http://www.cdc.gov/nchs/data/nvsr/nvsr53/nvsr53_20.pdf
Table provides
simulation results:
the proportion is far
from 0.512 over first
few weeks but in
long run settles
down around 0.512.
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Determining the Relative Frequency
Probability of an Outcome
Method 1: Make an Assumption about the Physical World
– Theoretical Probability
Example 7.2 A 3 number Lottery – player winds if his or her
three-digit number is chosen
1. What is the sample space?
2. What is the Theoretical Probability?
3. Does if mean you will win one time in
every thousand plays?
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Determining the Relative Frequency
Probability of an Outcome
Method 1: Make an Assumption about the Physical World
– Theoretical Probability
Example 7.3 Probability Alicia has to Answer a Question
There are 50 student names in a bag.
If names mixed well, can assume each
student is equally likely to be selected.
Probability Alicia will be selected to
answer the first question is 1/50 or .02.
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Determining the Relative Frequency
Probability of an Outcome
Method 2: Observe the Relative Frequency – Empirical
Probability
Example 7.4 The Probability of Lost Luggage
“3.91 per thousand passengers on U.S. airline
carriers will temporarily lose their luggage.”
Based on data collected over long run (a full year). Probability
a randomly selected passenger on a U.S. carrier will temporarily
lose luggage is 3.91/1000 = 1/256, or about 0.004.
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Proportions and Percentages
as Probabilities
Ways to express the relative frequency of lost luggage:
• The proportion of passengers who lose their
luggage is 1/256 or about 0.004.
• About 0.4% of passengers lose their luggage.
• The probability that a randomly selected
passenger will lose his/her luggage is about 0.004.
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Estimating Probabilities
from Observed Categorical Data
Assuming data are representative, the
probability of a particular outcome is
estimated to be the relative frequency
(proportion) with which that outcome
was observed.
Approximate margin of error
for the estimated probability is 1
n
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Example 7.5 Night-lights and Myopia
Revisited
Assuming these data are representative of a larger population,
what is the approximate probability that someone from that
population who sleeps with a nightlight in early childhood
will develop some degree of myopia?
Note: 72 + 7 = 79 of the 232 nightlight users developed some
degree of myopia. Estimated probability is 79/232 = 0.34.
Estimate based on sample of 232 with a margin of error of ~0.066.
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The Personal Probability Interpretation
Personal probability of an event = the degree
to which a given individual believes the event
will happen.
Sometimes subjective probability used because the
degree of belief may be different for each individual.
For example: the hiring of a particular person
Restrictions on personal probabilities:
• Must fall between 0 and 1 (or between 0 and 100%).
• Must be coherent.
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Homework
• Assignment:
• Chapter 7 – Exercise 7.5, 7.7, 7.10 and
7.12
• Reading:
• Chapter 7 – p. 221-228
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