Download Probability I

PHP 2510
Probability Part 1
Definition of probability measure
• set notation and arithmetic
• outcome, sample spaces, event
Probability measures
Important properties
• Complements
• Addition law
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Introduction
Propellers of early probability theories included gamblers in the
eighteenth century.
Founders of modern probability theories include Pierre-Simon
Laplace, Daniel Bernoulli, Carl F. Gauss, etc.
The theory of probability has broad relevance in many areas.
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Examples:
• Rhode Island reports 5 new cases of drug-resistant TB for the
first quarter of some year. Does this represent an outbreak or
epidemic?
• An individual tests positive for HIV on a rapid screening test.
What is the probability that the person actually is
HIV-positive?
• What is the likelihood of contracting HIV in one act of sexual
intercourse?
• What is the probability that someone with early stage breast
cancer will survive for 10 years? Does the probability differ if
she is 40 versus if she is 50?
• If a graduate program makes 10 offers of admission, what is the
expected number of students that will enroll? What is the
probability that 10 will enroll?
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• What is the probability of winning Powerball with the purchase
of one ticket?
• For a woman who is pregnant with triplets, what is the
probability of having three boys?
• A statistics professor claims that he can distinguish between
three varieties of cabernet. Given three unlabeled glasses, what
is the probability he can classify them correctly just by
guessing?
• In a room with 25 people, what is the probability that two
share the same birthday?
• Consider a batter whose average is .333 (gets a hit once in
every three at-bats). In a game where he has 4 at-bats, which
of the following is more likely: zero hits or more than 2 hits?
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A little mathematics...
A set is a collection of distinct objects (symbols, numbers, etc).
• E.g. {red, blue, green}, {1, 3, 5, 11}.
These distinct objects are called elements.
The order in which the elements of a set are listed is irrelevant.
• E.g. {red, blue, green} = {blue, green, red}.
Special sets: R = {all real numbers}; Z = {all integers}; N ={all
natural numbers}; ø is an empty set.
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Different ways to describe a set:
• List all elements.
E.g. {1, 3, 5, 2, 0}.
• Use “...”.
E.g. N = {1, 2, 3, ...}.
• Use “:”.
E.g. {m : m ∈ N and m > 8} is the same as {9, 10, 11, ...},
where the colon (“:”) means ”such that”.
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If every element of set A is also an element of set B, then A is said
to be a subset of B, written A ⊆ B.
• Graphically (Venn diagram),
If A ⊆ B and B ⊆ A, then A = B. It is true vice versa.
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Basic operations:
• The union of A and B, denoted by A ∪ B, is the set of all
things which are elements of either A or B.
Venn diagram,
• The intersection of A and B, denoted by A ∩ B, is the set of all
things which are elements of both A and B.
Venn diagram,
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• A universal set U is a set that contains (as elements) all the
entities one wishes to consider in a given situation.
• The complement of A, denoted by Ā or Ac , is the set of all
elements which are members of U , but not members of A
Venn diagram,
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Sample spaces
When thinking about the probability of observing a certain
outcome that occurs at random, the first step is to figure out
all possible outcomes.
The set of all possible outcomes is called sample space, which we
denote by Ω.
Sample space is analogue to universal set.
Example 1. Driving to work, you go through 3 intersections. At
each intersection you either stop (S) or continue (C). Sample
space is set of all possible outcomes
Ω = {SSS, SSC, SCS, CSS, SCC, CSC, CCS, CCC}
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Example 2. A woman gives birth to twins. Sample space of
possible gender combinations is
Ω = {F F, M F, F M, M M }
Example 3. A person attempts to quit smoking. The time from
the beginning of his quit attempt until potential relapse is all
times greater than zero.
Ω = {t : t ∈ R and t > 0}
Example 4. During the next year, you will have a certain number
of visits to the doctor. The sample space is
Ω = {0, 1, 2, . . .}
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Example 5. An individual takes a test for the HIV virus. The
test result is either positive or negative, and his true HIV
status is either HIV positive or HIV negative.
Ω = {(T + H + ), (T + H − ), (T − H + ), (T − H − )}
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Event
• Recall that sample space is all possible outcomes.
• An event is a collection of outcomes.
• In other words, an event is a subset of Ω.
• The empty set ø can be an event.
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Union. A ∪ B is the event that either A or B occurs.
Intersection. A ∩ B is the event that both A and B occur.
Example 6 (twins). A = “one boy”, B = “no boy”. Then
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Ω =
{F M, M F, M M, F F }
A
=
{F M, M F }
B
=
{F F }
A∪B
=
{F M, M F, F F }
A∩B
=
ø
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Example 7 (twins). A = “first is a boy”. B = “at least one boy”.
A
= {M M, M F }
B
= {M M, M F, F M }
A∪B
= {M M, M F, F M }
A∩B
= {M M, M F }
Example 8 (visits to the doctor). A = exactly 3 visits to the
doctor, B = 2 or more visits to the doctor.
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Ω =
{0, 1, 2, ...}
A
=
{3}
B
=
{2, 3, 4, . . .}
A∪B
=
{2, 3, 4, . . .}
A∩B
=
{3}
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Complement. Ac or A is the event that A does not occur.
Example 9 (twins). A = exactly one boy.
A
=
{F M, M F }
Ac
=
{M M, F F }
Example 10 (visits to the doctor). B = 2 or more visits to the
doctor.
B
=
{2, 3, 4, . . .}
Bc
=
{0, 1}
Properties of complements:
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A ∪ Ac
= Ω
A ∩ Ac
= ø
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Probability measure
Formally, it is a function labeled by P that maps subsets of the
sample space Ω to the [0,1] interval.
The probability measure P must satisfy the following axioms:
1. If an event A is in the sample space Ω, then P (A) ≥ 0
2. The probability of all possible events is one; i.e., P (Ω) = 1
3. If A and B are disjoint (cannot happen simultaneously;
A ∩ B = ø), then P (A ∪ B) = P (A) + P (B). This also holds
for more than two disjoint events.
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Two important properties
Complements: P (Ac ) = 1 − P (A)
Example 11: Suppose each combination of twins is equally likely
(probability 0.25). A = ‘two boys’. Then
A
= {M M }
P (A) =
Ac
P (Ac )
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0.25
= {M F, F M, F F }
=
1 − 0.25 = 0.75
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Addition Law: P (A ∪ B) = P (A) + P (B) − P (A ∩ B)
Example 12: Compute the probability that the first is a boy or
the second is a boy.
A
=
{M F, M M }
B
=
{F M, M M }
A∩B
=
{M M }
P (A ∪ B) =
P ({M F, M M }) + P ({F M, M M }) − P ({M M })
= 0.5 + 0.5 − 0.25
= 0.75
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Summary
Learning formal rules of probability is essential to understanding
randomness and, eventually, statistics
Probability measure maps subsets of a sample space to the unit
interval. That is, it converts events in a sample space to
probabilities that have values between 0 and 1. Become
familiar with probability axioms.
Properties of sets are important to computing probabilities
(unions, intersections, complements). Become familiar with
properties and laws governing sets – complement law, addition
law, etc.
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