Download 2. Probability

Chapter Goals
Explain basic probability concepts and
definitions.
2. Probability
Apply common rules of probability.
Compute conditional probabilities.
Determine whether events are statistically
independent.
©
Basic Terms
Random Experiment – a process leading to an
uncertain outcome
Basic Outcome – a possible outcome of a
random experiment
Sample Space – the collection of all possible
outcomes of a random experiment
Event – any subset of basic outcomes from the
sample space
Sample Space
The possible outcomes of a random
experiment are called the basic outcomes,
and the set of all basic outcomes is called
the sample space. The symbol S will be
used to denote the sample space.
Sample Space
- An Example -
What is the sample space for a roll
of a single six-sided dice?
S = [1, 2, 3, 4, 5, 6]
Mutually Exclusive
If the events A and B have no common basic
outcomes, they are mutually exclusive and their
intersection A ∩ B is said to be the empty set
indicating that A ∩ B cannot occur.
More generally, the K events E1, E2, . . . , EK
are said to be mutually exclusive if every pair of
them is a pair of mutually exclusive events.
Venn Diagrams
Venn Diagrams are drawings, usually
using geometric shapes, used to depict
basic concepts in set theory and the
outcomes of random experiments.
Intersection of Events A and B
S
S
A
A∩B
B
(a) A∩B is the striped area
A
(b) A and B are Mutually Exclusive
Collectively Exhaustive
Complement
Given the K events E1, E2, . . ., EK in the
sample space S. If E1 ∪ E2 ∪ . . . ∪EK = S,
these events are said to be collectively
exhaustive.
Let A be an event in the sample space S.
The set of basic outcomes of a random
experiment belonging to S but not to A
is called the complement of A and is
denoted by A.
Venn Diagram for the
Complement of Event A
B
Unions, Intersections, and
Complements
A die is rolled. Let A be the event “Number rolled is even”
and B be the event “Number rolled is at least 4.” Then
S
A = [2, 4, 6] and
A
A
B = [4, 5, 6]
A = [1, 3, 5] and B = [1, 2, 3]
A ∩ B = [4, 6]
A ∪ B = [2, 4, 5, 6]
A ∪ A = [1, 2, 3, 4, 5, 6] = S
Relative Frequency
Classical Probability
The relative frequency definition of
probability is the limit of the proportion of
times that an event A occurs in a large
number of trials, n,
The probability of an event A is
P(A) =
N A N(A)
=
N
N(S)
P(A) =
where NA is the number of outcomes that satisfy the
condition of event A and N is the total number of
outcomes in the sample space. The important idea here
is that one can develop a probability from fundamental
reasoning about the process.
Probability Postulates
where nA is the number of A outcomes and n
is the total number of trials or outcomes in
the population. The probability is the limit as
n becomes large.
Probability Rules
Let S denote the sample space of a random experiment, Oi,
the basic outcomes, and A, an event. For each event A
of the sample space S, we assume that a number P(A)
is defined and we have the postulates
1. If A is any event in the sample space S
0 ≤ P( A) ≤ 1
2.
nA
n
Let A be an event in S, and let Oi denote the basic
outcomes. Then
The Addition Rule of Probabilities:
Let A and B be two events. The probability
of their union is
P( A ∪ B ) = P( A) + P ( B) − P( A ∩ B)
P( A) = ∑ P(Oi )
A
where the notation implies that the summation
extends over all the basic outcomes in A.
3. P(S) = 1
Probability Rules
Venn Diagram for Addition Rule
P( A ∪ B ) = P( A) + P ( B) − P( A ∩ B)
Conditional Probability:
Let A and B be two events. The conditional probability
of event A, given that event B has occurred, is denoted
by the symbol P(A|B) and is found to be:
P(A∪B)
A
B
=
P(A)
A
P(B)
B
+
A
P(A∩B)
B
Probability Rules
-
A
P( A | B) =
B
provided that P(B > 0).
P( A ∩ B)
P( B)
Joint and
Marginal Probabilities
Joint and Marginal Probabilities
In the context of bivariate probabilities, the
intersection probabilities P(Ai ∩ Bj) are called joint
probabilities. The probabilities for individual
events P(Ai) and P(Bj) are called marginal
probabilities. Marginal probabilities are at the
margin of a bivariate table and can be computed
by summing the corresponding row or column.
The probability of a joint event, A ∩ B:
P(A ∩ B) =
number of outcomes satisfying A and B
total number of elementary outcomes
Computing a marginal probability:
P(A) = P(A ∩ B1 ) + P(A ∩ B 2 ) + L + P(A ∩ Bk )
where B1, B2, …, Bk are k mutually exclusive and
collectively exhaustive events
Ex. Weather Forecast
Joint Probability
Forecast
Outcome
Sunny
Rainy
Sunny
50
25
Forecast
Rainy
5
20
Outcome
Sunny
Rainy
Sunny
0.50
0.25
Rainy
0.05
0.20
Statistical Independence
Conditional Probability
Let A and B be two events. These events are said to be
statistically independent if and only if
P(A | B) = P(A)
Independence can be defined as
Forecast
Outcome
Sunny
Rainy
Sunny
0.67
0.33
(if P(B) > 0)
Rainy
0.20
0.80
P(B | A) = P(B)
(if P(A) > 0)
P ( A ∩ B ) = P( A) P( B )
More generally, the events E1, E2, . . ., Ek are mutually
statistically independent if and only if
P(E1 ∩ E 2 ∩ K ∩ E K ) = P(E1 ) P(E 2 ) K P(E K )
Ex. Stock Price
Conditional Probability
Forecast
Outcome
Up
Down
Up
30
30
Forecast
Down
20
20
Outcome
Up
0.50
0.50
Up
Down
NYSE
HK
Up
Down
Down
10
30
Overinvolvement Ratio
The probability of event A conditional on event B divided by
the probability of A conditional on activity C is defined as
the overinvolvement ratio:
P(A | B)
P(A | C)
An overinvolvement ratio greater than 1 implies that event A
increases the conditional odds ratio in favor of B:
P(B | A) P(B)
>
P(C | A) P(C)
P(B)
0.5
0.5
Odds
Ex. Spillover Effect
Up
40
20
Down
0.50
0.50
The odds in favor of a particular event are
given by the ratio of the probability of
the event divided by the probability of
its complement
The odds in favor of A are
odds =
P(A)
P(A)
=
1- P(A) P(A)
Example
Over-involvement Ratio =
P(NY=Up|HK=Up)/P(NY=Up|HK=Down)
= 0.8/0.4 = 2
Clearly,
P(HK=Up|NY=Up) > P(HK=Down|NY=Up).