IEOR 4106: Introduction to Operations Research: Stochastic Models
Spring 2011, Professor Whitt
Class Lecture Notes: Tuesday, January 18
Conditional Probability
We discussed several problems in Chapter 1 of Ross. I emphasized several key points:
1. Probability theory is a
branch of mathematics
, so it is important to pay attention to
deFnitions
and axioms (see the beginning of Section 1.3 in the Ross textbook).
You need to recall some elementary
set theory
. We use braces to denote a set, as appear
in the deFnition of the events
S
and
E
in Problem 1.18 below. Just to focus on one pedantic
(but useful) detail, note that
x
,
{
x
}
and
{{
x
}}
are di±erent objects:
x
is an element of the
set
{
x
}
, while
{
x
}
is an element of the set
{{
x
}}
;
x
is not an element of the set
{{
x
}}
. A set
containing the element
x
is not the same as the element
x
itself. It may help to Google “set
theory.” That is, look at Wikipedia, PlanetMath or Wolfram’s Mathworld. One or two pages
of reading should su²ce.
³ormally, a
probability measure
assigns probabilities to subsets of the sample space;
those subsets are called events. See Sections 1.11.3 of Ross.
2. We are focusing on
problem solving
. ³or that purpose, a good general strategy
is
divide and conquer
: break the problem into smaller pieces that are easier to analyze.
Skipping steps can cause errors.
3. It is helpful to draw
pictures
.
In particular, Chapter 1 emphasizes that a key idea overall is to remember and apply the
deFnition of
conditional probability
:
P
(
A

B
)
≡
P
(
AB
)
P
(
B
)
,
where
AB
≡
A
∩
B
denotes the
intersection
of the events (sets)
A
and
B
.
The following are exercises at the end of Chapter 1 in Ross.
1.18
(a) A family has two children. What is the probability that both are girls, given that
at least one is a girl? (Assume that each child is equally likely to be a boy or a girl.)
(b) Does the answer change if we rephrase the question: What is the probability that both
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 Spring '11
 WhitWard
 Operations Research, Conditional Probability, Probability, Probability theory, Probability space

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