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EAS31116/B9036: Statistics in Earth & Atmospheric Sciences
Lecture 1: Review of Probability
Instructor: Prof. Johnny Luo
www.sci.ccny.cuny.edu/~luo
Outlines
1. Definition of terms
2. Three Axioms of Probability
3. Some properties of probability
Probability deals with uncertainties
 When facing uncertainties, we need a way to describe it.
We can go with qualitative descriptors such as rain “likely”,
“unlikely” or “possible”.
 Probability is a quantitative way of expressing
uncertainty, e.g., 40% chance of rain.
 (Dictionary) Probability: the extent to which an event is
likely to occur, measured by the ratio of the favorable cases
to the whole number of cases possible.
 Probability builds upon an abstract mathematical system.
A Few Terms
 Events: A set of possible (uncertain) outcomes (e.g.,
flipping coin: you won’t know for sure which face will
come up).
 Sample Space (or Event Space): the set of all possible
events. Usually use capital letter S to represent it.
 Mutually Exclusive and Collectively Exhaustive (MECE)
events; ME: no more than one of the events can occur;
CE: at least one of the events will occur.
Venn Diagram
These are null space
Outlines
1. Definition of terms
2. Three Axioms of Probability
3. Some properties of probability
Axioms of Probability
Axiom: A self-evident truth that requires no proof; a
universally accepted principle; (mathematics) a proposition
that is assumed without proof for the sake of studying the
consequences that follow from it.
• For an event E in a sample space S
1. 0  P(E)  1
2. P( S )  1
3. P( E1  E2 )  P( E1 )  P( E2 ) where E1 and E2 mutually exclusive
The axioms are like the US Constitution. They are not very
informative about what probability exactly means or how
to estimate/interpret it.
There are two dominant views of the meaning of
probability: the Frequency view and the Bayesian view.
Frequency view: The true probability of of event {E} exists
and can be estimated through a long series of trials.
Bayesian view: There is no such a thing as true probability;
we just estimate it based on whatever information we
have in hand.
Here is how I would explain the basic difference to my grandma:
I have misplaced my phone somewhere in the home. I can use the phone
locator on the base of the instrument to locate the phone and when I press
the phone locator the phone starts beeping.
Problem: Which area of my home should I search?
Frequentist Reasoning:
I can hear the phone beeping. I also have a mental model which helps me
identify the area from which the sound is coming from. Therefore, upon
hearing the beep, I infer the area of my home I must search to locate the
phone.
Bayesian Reasoning:
I can hear the phone beeping. Now, apart from a mental model which helps
me identify the area from which the sound is coming from, I also know the
locations where I have misplaced the phone in the past. So, I combine my
inferences using the beeps and my prior information about the locations I
have misplaced the phone in the past to identify an area I must search to
locate the phone.
Outlines
1. Definition of terms & Venn Diagram
2. Three Axioms of Probability
3. Some properties of probability
Complement:
Complement:
Intersection (or joint probability)
Complement:
Intersection (or joint probability)
Union (one or the other, or both):
Complements of unions or intersections
Conditional Probability
Probability of an event, given that some other event has
occurred or will occur.
For example, the probability of freezing rain, given the
precipitation occurs.
Conditional Probability
Conditional probability can be defined in terms of the
intersection of the events of interest and the condition event.
Independence
Two events are independent if the occurrence or nonoccurrence of one does not affect the probability of the other.
Date (of Jan 1987); Precip (inch); T(max); T(min) in Ithaca NY
Estimate the probability
of at least 0.01 in. of
precipitation, given that
T(min) is at least 00F.
(14/31)/(24/31) = 14/24 = 0.58
Estimate the probability of at least 0.01 in. of
precipitation, given that T(min) is less than 00F.
(1/31)/(7/31) = 1/7 = 0.14
Think-Pair-Share: Why does wintertime
precipitation prefer higher T?
Law of Total Probability
Ei are a set of MECE events
Why do we bother?
Sometimes we only know the conditional probability of {A}
upon condition events Ei. The Law of Total Probability gives us
an opportunity to estimate the unconditional probability Pr{A}
Bayes’ Theorem
So, if we know conditional
probability Pr{E2|E1} and
unconditional probability
Pr{E1} and Pr{E2}, then we
can back out Pr{E1|E2} .
Bayes’ Theorem
So, if we know conditional
probability Pr{E2|E1} and
unconditional probability
Pr{E1} and Pr{E2}, then we
can back out Pr{E1|E2} .
=
Date (of Jan 1987); Precip (inch); T(max); T(min) in Ithaca NY
In previous example, we estimate the
conditional probability for precip occurrence
given T(min) above or below 00F. Now, let’s
use the Bayes’ Theorem to compute the
converse conditional probabilities,
concerning temperature events given that
preci did or did not occur.