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Statistics 510: Notes 4
Reading: Sections 2.4-2.5
I. Propositions about Probability Function Based on
Axioms (Section 2.4)
C
Proposition 4.1: P( E )  1  P( E ) .
C
Proof: Because E  E  S , by Axiom 2 we have
P( E  E C )  P( S )  1 .
Because E and E C are mutually exclusive, it follows from
Axiom 3 that
P( E  E C )  P ( E )  P ( E C ) .
C
C
Thus, P( E )  P( E  E )  P( E )  1  P( E ) .
Proposition 4.2: If E  F (meaning that every outcome in
E is contained in F ), then P( E )  P( F ) .
Proof: Note that the event F may be written in the form
F  E  (F  EC ) ,
where E and F  E C are mutually exclusive. Therefore, by
Axiom 3,
P( F )  P( E )  P( F  E C ) . By Axiom 1,
P( F  E C )  0 so that P( F )  P( E ) .
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Furthermore, from the proof of Proposition 4.2, we have
the difference rule that if E  F ,
P( F and not E )  P( F  E C )  P( F )  P( E ) .
Proposition 4.3: P( E  F )  P( E )  P( F )  P( E  F ) .
Proof: The Venn diagram suggests the statement of the
proposition is true. More formally, we have from Axiom 3
that
P( E )  P ( E  F C )  P ( E  F )
P( F )  P( E  F )  P ( E C  F )
P( E  F )  P( E  F C )  P ( E  F )  P ( E C  F )
From the first two equations, we have that
P( E  F C )  P( E )  P( E  F ),
P( E C  F )  P( F )  P( E  F )
Substituting these expressions in the expression for
P ( E  F ) , we conclude that
P( E  F )  P( E )  P( F )  P( E  F ) .
Note: Proposition 4.3 can be extended to provide an
expression for P( E1  E2   En ) ; see Proposition 4.4,
the inclusion-exclusion identity).
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Example 1: Winthrop, a premed student, has been
summarily rejected by all 126 U.S. medical schools.
Desperate, he sends his transcripts and MCATs to the two
least selective campuses he can think of, the two branch
campuses ( X and Y ) of Swampwater Tech. Based on the
success his friends have had there, he estimates that his
probability of being accepted at X is 0.7, and at Y , 0.4.
He also suspects that there is a 75% chance that at least one
of his applications will be rejected. What is the probability
that at least one of the schools will accept him?
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II. Sample Spaces Having Equally Likely Outcomes
For many experiments, it is natural to assume that all
outcomes in the sample space are equally likely to occur.
That is, consider an experiment whose sample space S is a
finite set, say S  {1, 2, , N } . Then it is often natural to
assume that
P({1})  P({2})   P({N }) .
This implies that
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P({i})  , i  1, , N
N
because P( S )  1 by Axiom 2 and
P( S )  P({1})   P({N }) by Axiom 3.
For a sample space with equally likely outcomes, for any
event E ,
number of outcomes in E
P( E ) 
(0.1)
number of outcomes in S
(this follows from Axiom 3).
In other words, if we assume that all outcomes of an
experiment are equally likely to occur, then the probability
of an event E equals the proportion of outcomes in the
sample space that are contained in E .
For counting the number of outcomes in E and the number
of outcomes in S , the methods of Chapter 1 are useful.
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Example 2: Your friend at Penn has a phone number that
starts with 498. Suppose that the remaining four digits in
your friend’s phone number are equally likely to be any
sequence. What is the probability that your friend’s phone
number contains seven distinct digits.
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Example 3: A poker hand consists of 5 cards. If the cards
have distinct consecutive values and are not all of the same
suit, we say that the hand is a straight. For instance, a hand
consisting of the five of spades, six of spades, seven of
spades, eight of spades and nine of hearts is a straight?
What is the probability that one is dealt a straight?
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Example 4: A deck of 52 playing cards is well shuffled and
the cards turned up one at a time until the first ace appears.
Is the next card – that is, the card following the first ace –
more likely to be the ace of spades or the two of clubs?
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Example 5: A football team consists of 20 offensive and 20
defensive players. The players are to be paired in groups of
2 for the purpose of determining roommates. If the pairing
is done at random, what is the probability that there are no
offensive-defensive roommate pairs? What is the
probability that there are 2i offensive-defensive roommate
pairs, i=1,2,...,10?
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