Download probability models: finitely many outcomes

CHAPTER 4
SECTIONS 4.1 – 4.2
PROBABILITY
AND
PROBABILITY RULES
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PROBABILITY MODELS: FINITELY
MANY OUTCOMES
DEFINITION:
PROBABILITY IS THE STUDY OF
RANDOM OR NONDETERMINISTIC
EXPERIMENTS. IT MEASURES THE
NATURE OF UNCERTAINTY.
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PROBABILISTIC TERMINOLOGIES
• RANDOM EXPERIMENT
AN EXPERIMENT IN WHICH ALL
OUTCOMES (RESULTS) ARE KNOWN
BUT SPECIFIC OBSERVATIONS
CANNOT BE KNOWN IN ADVANCE.
EXAMPLES:
• TOSS A COIN
• ROLL A DIE
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SAMPLE SPACE
•
THE SET OF ALL POSSIBLE OUTCOMES OF
A RANDOM EXPERIMENT IS CALLED THE
SAMPLE SPACE.
•
NOTATION:
•
EXAMPLE
S
1. FLIP A COIN THREE TIMES
S=
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EXAMPLE
• AN EXPERIMENT CONSISTS OF FLIPPING A
COIN AND THEN FLIPPING IT A SECOND TIME
IF A HEAD OCCURS. OTHERWISE, ROLL A
DIE.
• RANDOM VARIABLE
THE OUTCOME OF AN EXPERIMENT IS
CALLED A RANDOM VARIABLE. IT CAN ALSO
BE DEFINED AS A QUANTITY THAT CAN TAKE
ON DIFFERENT VALUES.
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EXAMPLE
• FLIP A COIN THREE TIMES. IF X
DENOTES THE OUTCOMES OF THE
THREE FLIPS, THEN X IS A RANDOM
VARIABLE AND THE SAMPLE SPACE IS
•S=
{HHH,HHT,HTH,THH,HTT,THT,TTH,TTT}
• IF Y DENOTES THE NUMBER OF HEADS
IN THREE FLIPS, THEN Y IS A RANDOM
VARIABLE. Y = {0, 1, 2, 3}
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EVENTS
• AN EVENT IS A SUBSET OF A SAMPLE
SPACE, THAT IS, A COLLECTION OF
OUTCOMES FROM THE SAMPLE SPACE.
• EVENTS ARE DENOTED BY UPPER CASE
LETTERS, FOR EXAMPLE, A, B, C, D.
• LET E BE AN EVENT. THEN THE
PROBABILITY OF E, DENOTED P(E), IS
GIVEN BY
n
P ( E )   P ( wi )
i 1
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Probability of an Event, E
• LET E BE ANY EVENT AND S THE
SAMPLE SPACE. THE PROBABILITY OF
E, DENOTED P(E) IS COMPUTED AS
n( E )
P( E ) 
n( S )
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Probability of an Event
• The probability of an event A, denoted
by P(A), is obtained by adding the
probabilities of the individual outcomes
in the event.
•
When all the possible outcomes are
equally likely,
number of outcomes in event A
P( A) 
number of outcomes in the sample space
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Probability Model
• A probability model is a mathematical
description of long-run regularity consisting
of a sample space S and a way of assigning
probabilities to events.
• ILLUSTRATION: Tossing of a coin
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Probability Rules
• We can use graphs to represent probabilities and
probability rules.
• Venn diagram:
S
A
Area = Probability
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Probability Rules
• Rule 1:
– A probability is a number between 0 and 1.
For any event A, 0 ≤ P(A) ≤ 1.
– A probability can be interpreted as the
proportion of times that a certain event
can be expected to occur.
– If the probability of an event is more than
1, then it will occur more than 100% of the
time (Impossible!).
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Probability Rules
• Rule 2:
– All possible outcomes together must have
probability 1. P(S) = 1.
– Because some outcome must occur on
every trial, the sum of the probabilities for
all possible outcomes must be exactly
one.
– If the sum of all of the probabilities is less
than one or greater than one, then the
resulting probability model will be
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incoherent.
Illustration of Rule 2
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EXAMPLES
1. A COIN IS WEIGHTED SO THAT HEADS IS
TWICE AS LIKELY TO APPEAR AS TAILS.
FIND P(T) AND P(H).
2. THREE STUDENTS A, B AND C ARE IN A
SWIMMING RACE. A AND B HAVE THE
SAME PROBABILITY OF WINNING AND
EACH IS TWICE AS LIKELY TO WIN AS C.
FIND THE PROBABILITY THAT B OR C
WINS.
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Probability Rules
• Rule 3 (Complement Rule):
– The probability that an event does not occur is 1
minus the probability that the event does occur.
– The set of outcomes that are not in the event A is
called the complement of A, and is denoted by AC.
– P(AC) = 1 – P(A).
– Example: what is the probability of getting at least
1 head in the experiment of tossing a fair coin 3
times?
HHH
HHT
HTH
HTT
THH
THT
TTH
TTT
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COMPLEMENT OF AN EVENT
• THE COMPLEMENT OF AN EVENT A WITH
RESPECT TO S IS THE SUBSET OF ALL
ELEMENTS(OUTCOMES) THAT ARE NOT IN
A.
C
A
• NOTATION:
OR
'
A
P( A )  1  P( A)
C
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Probability Rules
• Complement rule: P(AC) = 1 – P(A).
AC
A
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Complement of an Event, A
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Probability Rules
• Rule 4 (Multiplication Rule):
– Two events are said to be independent if
the occurrence of one event does not
influence the occurrence of the other.
– For two independent events A and B, the
probability that both A and B occur is the
product of the probabilities of the two
events.
– P(A and B) = P(A) X P(B).
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Examples From Handout
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Combining Events
• INTERSECTION OF EVENTS
THE INTERSECTION OF TWO EVENTS A AND
B, DENOTED
A B
IS THE EVENT
CONTAINING ALL ELEMENTS(OUTCOMES)
THAT ARE COMMON TO A AND B.
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UNION OF EVENTS
• THE UNION OF TWO EVENTS A AND B,
DENOTED,
A B
IS THE EVENT CONTAINING ALL THE
ELEMENTS THAT BELONG TO A OR B OR
BOTH.
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Intersection and Union of Events
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MUTUALLY EXCLUSIVE(DISJOINT) EVENTS
• TWO EVENTS A AND B ARE MUTUALLY
EXCLUSIVE(DISJOINT) IF
A B  
THAT IS, A AND B HAVE NO OUTCOMES IN
COMMON.
IF A AND B ARE DISJOINT(MUTUALLY
EXCLUSIVE),
P ( A  B )  P ( )  0
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Probability Rules
• Rule 5 (Addition Rule):
– If two events have no outcomes in
common, they are said to be disjoint (or
mutually exclusive).
– The probability that one or the other of two
disjoint events occurs is the sum of their
individual probabilities.
P(A or B) = P(A) + P(B), provided that A
and B are disjoint.
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Addition Rule
• Addition rule: P(A or B) = P(A) + P(B), A
and B are disjoint.
S
B
A
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Example From Handout
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Probability Rule 6 – The General
Addition Rule
• For two events A and B, the probability
that A or B occurs equals the
probability that A occurs plus the
probability that B occurs minus the
probability that A and B both occur.
P(A or B) = P(A) + P(B) – P(A and B).
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Probability of the Union of
Events
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Examples From Handout
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Solution
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NULL EVENT: AN EVENT THAT HAS NO
CHANCE OF OCCURING. THE
PROBABILITY OF A NULL EVENT IS
ZERO.
P( NULL EVENT ) = 0
• CERTAIN OR SURE EVENT: AN
EVENT THAT IS SURE TO OCCUR.
THE PROBABILITY OF A SURE OR
CERTAIN EVENT IS ONE.
P(S) = 1
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Extra-Credit – Question 1
• In a large Statistics lecture, the professor reports that 52% of
the students enrolled have never taken a Calculus course, 34%
have taken only one semester of Calculus, and the rest have
taken two or more semesters of Calculus. The professor
randomly assigns students to groups of three to work on a
project for the course.
(i) What is the probability that the first group member you
meet has studied some Calculus?
(ii) What is the probability that the first group member you
meet has studied no more than one semester of Calculus?
(iii) What is the probability that both of your two group
members have studied exactly one semester of Calculus?
(iv) What is the probability that at least one of your group
members has had more than one semester of Calculus?
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Extra Credit – Questions 2 and 3
Question 2
A PAIR OF FAIR DICE IS TOSSED. FIND THE
PROBABILITY THAT THE MAXIMUM OF THE TWO
NUMBERS IS GREATER THAN 4.
Question 3
ONE CARD IS SELECTED AT RANDOM FROM 50
CARDS NUMBERED 1 TO 50. FIND THE PROBABILITY
THAT THE NUMBER ON THE CARD IS (I) DIVISIBLE BY
5, (II) PRIME, (III) ENDS IN THE DIGIT 2.
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