MATH 141-501
Section 7.6 Lecture Notes
A priori probability – the likelihood that an event will occur (calculated
before experiment)
A posterior probability – the likelihood that an event did occur (calculated after observing the outcomes of the experiment
Example: (from textbook) A card is drawn from a well-shuffled, standard,
52-card deck, then another card is drawn. Find the probability that...
...the first card is a club, given that the second heart is a club?
...the first card is a diamond, given that the second card is a diamond?
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Example - Information in Tree Diagrams
A tree diagram for a two-stage experiment is given below.
Fill in the missing information and use the diagram to find...
...P (A) · P (D | A)
...P (A) · P (D | Ac )
...P (D)
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Bayes’ Theorem
Bayes’ Theorem is a tool to solve problems about multi-stage experiments
without having to draw tree diagrams. In order to use Bayes’ Theorem, we need
to have some events A1 , A2 , . . . , An so that we are certain that exactly one
of the events happened.
In “math speak”, this would be expressed as:
1. Ai ∩ Aj = ∅ if i neqj (the events in the collection are mutually exclusive)
S
S
2. A1 A2 . . . An = S (we know with absolute certainty that one of the
events happened)
Definition: A collection of events satisfying the properties above is called
a partition of the sample space.
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Formal Statement of Bayes’ Theorem
Bayes’ Theorem: When A1 , A2 , . . . , An is a partition of the sample space
S, and E is an event of the experiment such that P (E) 6= 0 and P (Ai ) 6= 0 for
1 ≤ i ≤ n.
Then for each i between 1 and n, the aposteriori probability P (Ai | E) is given
by
P (Ai | E) =
=
P (Ai ) · P (E | Ai )
P (A1 ) · P (E | A1 ) + P (A2 ) · P (E | A2 ) + · · · + P (An ) · P (E | An )
product of probabilities of branches to E which go through Ai
sum of product of probabilities of ALL branches leading to E
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Example
The Venn diagram below represents a sample space S in which the events
A, B, and C are a partition of S.
Find...
... P (D)
...P (A | D)
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Example
The Venn diagram below represents a sample space S in which the events
A, B, and C are a partition of S.
Using the information in the diagram, draw a tree diagram illustrating the
probabilities of the events A, B, C, and D,.
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Example - Drug Tests
A blood test is used to test the presence of performance enhancing drugs
(PEDs) in athletes. If the athlete recently used PEDs , there is a 99% chance
that the test will detect it. However, there is a 0.9% chance of a false positive
(the athlete tests positive even though they did not use PEDs). It is believed
that about 50% of all professional athletes use PEDs.
If an athlete tests positive, what is the probability that the athlete actually
used PEDs?
What is the probability that an athlete did not use PEDs, if they tested
positive twice? (Assume the results of the successive tests are independent.)
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Example - DWI
Data compiled by a certain town’s police department shows that on a given
night, it is estimated that 0.5% of all drivers are impaired. Among drivers who
are impaired, it is estimated that roughly 50% will be stopped for a moving
violation. Among drivers who are not impaired, roughly 1% will be stopped for
a moving violation.
Make a table to represent the given information.
Question: IF a driver in that town is stopped for a moving violation, what
is the probability that he or she ...
...will be impaired?
...will not be impaired?
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Example - 2012 Election
In the 2012 Presidential Election, Barack Obama gained 60% of the vote
among 18-29 year olds, 52% of the vote among 30-44 year olds, 47
...an 18-29 year old?
...a 30-44 year old?
...someone 45 years of age or older?
Source:
http://www.ropercenter.uconn.edu/elections/how_groups_voted/voted_12.html
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Example - Drawing Cards ( based on an example by Joe Kahlig )
An experiment is performed where cards are drawn successively from a standard, well-shuffled deck.
Find the probability that...
... the second card drawn is a spade.
... the third card drawn is a spade.
...the 30th card drawn is a spade.
Example - Independent Events
If a fair coin is flipped ten times in a row, what is the probability that...
...the tenth toss is heads?
... the tenth toss is heads, given that the first nine tosses were heads?
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