Download probability of the event A

When you encounter a challenge, just do your best,
then accept the results and finally let it go.
Probability
Basic
terms
Conditional probability
Bayes’ theorem
Application
2
Random Circumstance

3
A random circumstance is one in which the
outcome is unpredictable. The outcome is
unknown until we observe it.
Basic Terms
4

Sample space S: the collection of all possible
outcomes of a random circumstance

An event is a collection of one or more outcomes in
the sample space. The event A is said to “occur” if
the outcome falls in the set A. So each time an
outcome is observed, a given event A either occurs
or does not occur.
Definition of Probability
5

P(A) = the probability of the event A
= the long-run relative frequency of occurrence of the event A
= the proportion or fraction of times that the event A occurs in a
large number of independent identical random circumstances.

P(A) can also represent the likelihood of occurrence of A based on
a subjective assessment. (eg. What is the probability that a flipped
coin shows heads up?)
Basic Probability Rules
6

Between 0 and 1

The sum of the probabilities over all possible
outcomes is 1

The sum of the probabilities of simple events
in the event
Equally Likely Probability Model

If the sample space S is finite in number and
the outcomes have the same likelihood of
occurrence, then each outcome has
probability equal to 1 divided by the number
of possible outcomes and so
# in A
P ( A) 
.
# in S
7
Basic Set Theory Terms
A or B = AUB = the union of A and B = the set of all
outcomes that are in A or in B or in both A and B.
 A and B = A∩B = the intersection of A and B = the set of
all outcomes that are in both A and B.
 Not A = Ac = the complement of A = the set of all
outcomes that fall outside of A.
 A and B are disjoint or mutually exclusive if is the empty
set; that is, A and B have no outcomes in common.
Therefore, A and B are mutually exclusive if and only if
they cannot occur together.
* Show Venn diagram

8
More Probability Rules



9
Addition rule:
P(A U B) = P(A) + P(B) - P(A and B)
P(A1 U A2 U…U Ak)=P(A1)+P(A2)+…+P(Ak)
if A1, A2, …, Ak are mutually exclusive events
(no two events can occur together).
Complement rule:
P(Ac) = 1-P(A)
Conditional Probability

10
When we calculate P(A), the probability of an event A, we
assume that we know nothing about the outcome of the random
experiment except that it will be a member of the sample space.
But when we calculate P(A | B), the conditional probability of an
event A given the event B, we assume that the outcome will be
a member of the event B. That is, we are given some partial
information about the outcome (it will fall in B) and we have to
calculate the probability that the outcome will also fall in A.
Sometimes, P(A | B) can be calculated using our intuition. In
other situations, we may have to use the formula:
P( A  B)
P( A | B) 
P( B)
Independent Events

Knowing that B will occur may or may not affect the
probability of A. If the occurrence of B does not
change the probability of A, then A and B are said
to be independent; otherwise, they are dependent.
So A and B are independent if and only if
P(A|B)=P(A)
Note that P(A|B)=P(A) is equivalent to P(B|A)=P(B).

11
More Probability Rules
12

Multiplication rule:
P(A ∩ B) = P(A) x P(B|A) = P(B) x P(A|B)

P(A1 ∩ A2 ∩…∩ Ak)=P(A1)xP(A2)x…xP(Ak)
if A1, A2, …, Ak are independent events
Tools for Finding Probabilities
13

When conditional or joint probabilities are
known for two events  Two-way tables

For a sequence of events, when conditional
probabilities for events based on previous
events are known  Tree diagrams
Total Probability Law
•Assume the sample space S is “partitioned” by the events A1,
A2, …, Ak meaning that
1) S  A1 A2 ...  Ak and
2)
Ai  Aj  
if i ≠ j (the events are pairwise disjoint).
This says that the outcome of the random circumstance will be in
one and only one of the partitioning events.
•Total Probability Law asserts that for any event B,
P( B)  P( B | A1) P( A1)  ...  P( B | Ak ) P( Ak )
14
Bayes’ Theorem
P( Ai  B)
P( Ai | B) 
P( B | A1) P( A1)  ...  P( B | Ak ) P( Ak )
Think of the events A1, A2,…, Ak as representing all possible
conditions that can produce the observable “effect” B. In this
context, the probabilities P(Ai)’s are called prior probabilities. Now
suppose that the effect B is observed to occur. Bayes’ theorem
gives a way to calculate the probability that B was produced or
caused by the particular condition Ai than by any of the other
conditions. The conditional probability P(Ai|B) is called the posterior
probability of Ai .
15
Example: Hearing Impairments
Employment
Status
Currently employed
Currently
unemployed
Not in the labor force
Total
16
Population size
Impairments
98,917
552
7462
27
56,778
368
163,157
947
Diagnostic Tests




17
A diagnostic testing or screening is the
application of a test to individuals who have not
yet exhibited any clinical symptoms in order to
classify them with respect to their probability of
having a particular disease.
“sensitivity” is the true positive rate
“specificity” is the true negative rate
“prevalence” is the proportion of subjects with
the disease in a population
Example: Pap Smear
1,000,000
women
Cervical
cancer
83
Test +
70
18
No cervical
cancer
999,917
Test –
13
Test +
186,385
Test –
813,532
Relative Risk

The relative risk, RR, is the chance that a
subject receiving some exposure will develop
disease relative to the chance that an
unexposed subject will develop the same
disease.
P(disease | exposed )
RR 
P(disease | unexposed)
19
Odds and Odds Ratio


If an event (disease) takes place with chance
p, the odds of the event are p/(1-p), the ratio
of the chance the event occurs to the chance
it does not occur.
The odds ratio (OR) of a disease for exposed
subjects to unexposed subjects is:
P(disease | exposed ) /[1  P(disease | exposed )]
OR 
P(disease | unexposed) /[1 - P(disease | unexposed) ]
20
Example: Smoke vs. Lung Cancer

In US, the chance that a man over the age of
35 dies of lung cancer is .002679 for current
smokers and .000154 for nonsmokers

RR=?
OR=?

21