Transcript Basic Concepts of Discrete Probability

Basic Concepts of Discrete
Probability
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Sample Space
• When “probability” is applied to something,
we usually mean an experiment with certain
outcomes.
• An outcome is any one of the possibilities that
may be expected from the experiment.
• The totality of all these outcomes forms a
universal set which is called the sample space.
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Sample Space
• For example, if we checked occasionally the
number of people in this classroom on
Wednesday from 11am to 12-15pm, we
should consider this an experiment having 19
possible outcomes {0,1,2,…,13,19} that form a
universal set.
• 0 – nobody is in the classroom, … 19 – all
students taking the Discrete Mathematics
Class and the instructor are in the classroom
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Sample Space
• A sample space containing at most a
denumerable number of elements is called
discrete.
• A sample space containing a nondenumerable
number of elements is called continuous.
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Sample Space
• A subset of a sample space containing any
number of elements (outcomes) is called an
event.
• Null event is an empty subset. It represents an
event that is impossible.
• An event containing all sample points is an
event that is certain to occur.
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Sample Space
We toss a single die, what are the possible outcomes,
which form the sample space?
{1,2,3,4,5,6}
We toss a pair of dice, what is the sample space?
Depends on what we’re going to ask.
Often convenient to choose a sample
space of equally likely events.
{(1,1),(1,2),(1,3),…,(6,6)}
Sample Space
• The following sets are subsets of the sampling
set {1, 2, 3, 4, 5, 6} in the die-tossing
experiments and therefore they are the
events:
• A={1, 2, 4, 6}
• B={n: n is an integer and 4  n  6 }
• C={n: n is an even positive integer less than 7}
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The Probability
• The classical definition given by Laplace says that
the probability is the ratio of the number of
favorable events to the total number of possible
events.
• All events in this definition are considered to be
equally likely: e.g., throwing of a true die by an
honest person under prescribed circumstances…
• …but not checking the number of people in the
classroom.
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The Probability
• According to the Laplace definition, for any
event E in a finite sample space S (recall that if
E is an event then E  S ) consisting of
equally likely outcomes, the probability of E,
which is denoted P(E) is
|E|
P( E ) 
|S|
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The Probability
• The following properties are important:
0 | E | M , 0 | S | N
|E|
0
 1  0  P( E )  1
|S|
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The Probability
• The following properties are important:
If P( E )  p  P( E )  1  p
P( E
E)  1
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Die-tossing experiments
• Let us find the probabilities of the following
events in the die-tossing experiments.
• The sampling space is S={1, 2, 3, 4, 5, 6}
• A={1, 2, 4, 6} P(A)=|A|/|S|=4/6=2/3
• B={n: n is an integer and 4  n  6 }
P(B)=|B|/|S|=2/6=1/3
• C={n: n is an even positive integer less than 7}
P(C)= |C|/|S|=3/6=1/2
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Coin experiment
• Let us flip a properly balanced coin three
times. What is the probability of obtaining
exactly two heads?
• Each flip of the coin has two possible results
(H) or (T) => according to the multiplication
principle there are 2x2x2=8 possible outcomes
for 3 flips S={HHH, HHT, HTH, HTT, THH, THT,
TTH, TTT} , three of which are favorable
E={HHT, HTH, THH} => P(E)=|E|/|S|=3/8
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Card experiment
What is the probability that a 5 card poker hand contains a royal
flush?
S = all 5 card poker hands.
A = all royal flushes
P(A) = |A|/|S|
|A|=4
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|S|= C52  C (52,5)
P(A) = 4/C(52,5)
“Pen” experiment
• Suppose that there are 2 defective pens in a
box of 12 pens. If we choose 3 pens in
random, what is the probability that we do
not select a defective pen?
• The sample space S consists of all possible
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selections of 3 pens chosen from 12: C12  C(12,3)
• The favorable event E is to chose 3 pens
among 10 nondefective ones C103  C(10,3)
C103 C (10,3)  10!   12!  120 6
• P(E)=|E|/|S|= 3 


/

C12
C (12,3)
 7!3!   9!3! 
220
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Homework
• Read Section 8.5 paying a closer attention to
examples.
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