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Computing Fundamentals 2
Lecture 6
Probability
Lecturer: Patrick Browne
http://www.comp.dit.ie/pbrowne/
Probability
• If a die is thrown we consider it certain that
it will land, with a random chance that it
will show a 6. With s successes out of n
experiments f=s/n is called the relative
frequency of success. It becomes stable in
the long run. It is this long term stability
(limit) that forms the basis of probability.
Sample Space and Events
• The sample space S is the set of all
possible outcomes of a given experiment.
• An element or outcome in S is called a
sample point (or sample).
• An event A is a set of outcomes, it is a
subset of the sample space.
• The singleton {a} where a  S is called
an elementary event.
• The empty set, , sometimes represents
an impossible event.
Sample Space and Events
• An event gives rise to a set hence we can use
set operations to combine events.
• A  B is the event that occurs whenever A
occurs or B occurs (or both)
• A  B is the event that occurs whenever A and
B both occur.
• Ac is the event that occurs whenever A does not
occur (called the complement of A)
• Two events are mutually exclusive if they are
disjoint: A  B = .
Sample Space and Events
• Toss a die and observe the top number
S={1,2,3,4,5,6}
• A even number event, B odd number
event, C prime number event.
• A={2,4,6} B={1,3,5} C={2,3,5}
• A  C ={2,3,4,5,6}
• B  C = {3,5}
• Cc = {1,4,6} (1 is not prime)
• A and B are mutually exclusive.
Sample Space and Events
• Toss 3 coins and observe the H & T
sequence
• S={HHH,HHT,HTH,HTT,THH,THT,TTH,TTT}
• Let A be the two consecutive heads event,
B same outcome event.
• A={HHH,HHT,THH} B={HHH,TTT}
• A  B = {HHH} is the elementary
event with only heads.
Probability Spaces
• A probability space is a triple (S, A, P),
where:
• S sample space, all possible outcomes.
• A event space, sample events/outcomes
• P is a probability measure.
• We also have a set of probability axioms
e.g. the probability of an event is a nonnegative real number.
Probability Spaces
• A probability space consists of a sample
space together with a positive, additive
measure, called a probability measure,
which sums to one; the points of the
sample space represent the different
possible outcomes of the phenomenon,
and the probability measure assigns
probabilities to sets of outcomes.
Finite Probability Spaces
• Let S be a finite sample space
S={a1,a2,a3...an}. A finite probability space
is obtained by assigning to each sample point
aiS a real number pi, called the probability of
ai satisfying the following conditions:
– Each pi is non-negative.
– The sum of pi is one.
• We write P(A) for the sum of the probabilities
sample points in A.
Finite Probability Spaces
• Three runners A,B,C; A is twice a likely to win
as B, and B is twice as likely to win as C. What is
P(A),P(B),P(C) winning?
• Let P(C) = p
• P(B) = 2p
• P(A)=4p
• p+2p+4p=1 therefore p = 1/7
• P(A)=4/7, P(B)=2/7, P(C)=1/7
• P({B,C}) = P(B)+P(C)=3/7
Equiprobable Spaces
• If all the sample points within a given finite
probability space are equal to each other,
then it is known as an equiprobable space.
An example would be a fair die, where
each number is equally possible
P(1) = P(2) = P(3) = P(4) =P(5) = P(6) = 1/6
Equiprobable Spaces
• If S contains n points, then the probability
of each point is 1/n. If an event A contains
r points then its probability is:
r  1/n = r/n
• P(A) = number of elements in A
number of elements in S
Equiprobable Spaces
• S = cards in the deck = 52
• A = card is spade
• B = card is a face
• P(A) = 13/52
• P(B) = 12/52
• P(AB) = 3/52
Axioms of Finite Probability Spaces
1. For every event A, 0P(A)1
2. P(S)=1, where S is sample space,
3. If events A and B are mutually exclusive
(or disjoint), then
P(AB) = P(A) + P(B)
Theorems of Finite Probability
Spaces
1. P() = 0
2. P(Ac)= 1 – P(A)
3. P(A\B)= P(A) - P(AB)
4. AB implies P(A)P(B)
5. P(A)  1
6. P(AB) = P(A) + P(B) - P(AB)
7. P(AB) = P(A) × P(B|A)
Where P(B|A) reads the probability B given A
Addition
P(AB) = P(A) + P(B) - P(AB)
• Sums are used when we have two events,
and we want to know the probability that
either event occurs (Event A union Event B).
In the Addition Rule, A and B may or may
not be disjoint. Mutually exclusive or disjoint
events cannot occur together, so we have:
P(A ⋂ B) = 0.
• Then the addition rule reduces to:
P(A U B) = P(A) + P(B)
Addition Rule Example
•
•
•
•
•
Suppose a student is selected at random from
100 students where 30 are taking maths, 20
are taking chemistry, and 10 are taking maths
and chemistry. Find the probability p that the
student is taking maths or chemistry.
P(M) = 30/100, P(C)=20/100
P(M  C) = 10/100
P(MC)=P(M) + P(C) - P(MC)
P(MC)= 30/100+20/100–10/100=2/5
Rule of Multiplication
• Is used when we want to know the
probability that two events occur (Event A
intersection Event B).
• Rule of Multiplication The probability that
Events A and B both occur is equal to the
probability that Event A occurs times the
probability that Event B occurs, given that
A has occurred.
• P(AB) = P(A) × P(B|A)
Rule of Multiplication
• A bag contains 6 red marbles and 4 blue
marbles. Two marbles are drawn without
replacement from the bag. What is the
probability that both of the marbles are blue?
• A = first marble is blue, B = second marble is
blue.
• Therefore, P(A) = 4/10, P(B|A) = 3/9.
• Using P(A ∩ B) = P(A) P(B|A)
P(A ∩ B) = (4/10) * (3/9) = 12/90 = 2/15
Rule of Multiplication
• A bag contains 6 red marbles and 4 blue
marbles. Two marbles are drawn with
replacement from the bag. What is the
probability that both of the marbles are blue?
• A = first marble is blue, B = second marble is
blue.
• Therefore, P(A) = 4/10, P(B|A) = 4/10.
• Using P(A ∩ B) = P(A) P(B|A)
P(A ∩ B) = (4/10) * (4/10) = 16/100 = 4/25
Conditional Probability
• E is an event in S with P(E)>0.
Conditional probability of A is defined as
the probability that A has occurred after E
has occurred. We say the conditional
probability of A given E:
• P(A|E) = P(AE)
P(E)
• P(A|E) = number of elements in AE
number of elements in E
Example: Conditional Probability
• Alternatively
• P(A|E) = number of ways A and E can occur
number of ways E can occur
• Given the sum of a pair of tossed die is 6.
• E={sum is 6},5 ways =
{(1,5),(2,4),(3,3),(4,2),(5,1)}
• A= {has at least one two},2 ways=
{(2,4),(4,2)}
• P(A|E)=2/5
Example 2: Conditional Probability
P(A|E) = number of ways A and E can occur
number of ways E can occur
• From a class has 12 boys and 4 girls,
3 students are selected. What is the
probability that they are all boys?
• P=Comb(12,3)/Comb(16,3)=11/28
• Alternatively
• P=(12/16)(11/15)(10/14) = 11/28
Independence
• Two events are independent if the occurrence of
one of the events gives us no information about
whether or not the other event will occur; that is,
the events have no influence on each other.
• We say that two events, A and B, are
independent if the probability that they both
occur is equal to the product of the probabilities
of the two individual events, i.e.
• P(AB) = P(A)  P(B)
• If two events are independent then they cannot
be mutually exclusive (disjoint) and vice versa.
Example: Independence
• Events A and B are independent if
• P(A∩B) = P(A)
∙ P(B)
• otherwise they are dependent.
• A coin tossed three times:
•
•
•
•
S={HHH,HHT,HTH,HTT,THH,THT,TTH,TTT}
A={first toss head}
B={second toss head}
C={exactly 2 heads tossed in a row}
Example: Independence
• Continuing, coin tossed three times:
•
•
•
•
•
•
P(A)={HHH,HHT,HTH,HTT}=4/8 (1st head)
P(B)={HHH,HHT,THH,THT}=4/8 (2nd head)
P(C)={HHT,THH} = 1/4 (2 heads in row)
P(AB)=P({HHH,HHT})= 1/4
P(AC)=P({HHT})= 1/8
P(BC)=P({HHT,THH})= 1/4
Example: Independence
• Continuing, coin tossed three times:
• P(AB)=P({HHH,HHT})= 1/4
• P(AC)=P({HHT})= 1/8
• P(BC)=P({HHT,THH})= 1/4
• P(A)P(B)=(1/2)(1/2)=(1/4)= P(AB)
• P(A)P(C)=(1/2)(1/4)=(1/8)= P(AC)
• P(B)P(C)=(1/2)(1/4)=(1/8) P(BC)
Not independent, B and C are dependent.
Repeated Trials
• The Law of Averages states, in the long
run, over repeated trials, random
fluctuations eventually average out and
the average of our observations will
approach the expected value. But at the
same time with increasing numbers of
observations, the number of observations
that differ from what we expect will be
larger.
Repeated Trials
• Let S* be a finite probability space. By n
independent or repeated trials we mean
the probability space S consisting of all
ordered n-tuples of elements of S*, with
the probability of n-tuple defined to be the
product of the probabilities of its
components.
• P(s1,s2,s3...sn)=P(s1)P(s2) 
P(sn)
Repeated Trials
• Let probability space S*={P(a),p(b),P(c)}
represents probabilities three runners
winning a race. Their probabilities of
winning are P(a)=1/2, P(b)=1/3,
P(c)=1/6.
• If there are two races then the sample
space S consisting of two repeated trials
is:
• S={aa,ab,ac,ba,bb,bc,ca,cb,cc}
Repeated Trials
• S={aa,ab,ac,ba,bb,bc,ca,cb,cc}
• The probability of the sample points of S are:
P(aa)=(1/2)(1/2)=1/4
P(ab)=(1/2)(1/3)=1/6
P(ac)=(1/2)(1/6)=1/12
P(ba)=(1/3)(1/2)=1/6
P(bb)=(1/3)(1/3)=1/9
P(bc)=(1/3)(1/6)=1/18
P(ca)=(1/6)(1/2)=1/12
P(cb)=(1/6)(1/3)=1/18
P(cc)=(1/6)(1/6)=1/36
• The probability of c winning first race and a the
second is P(ca)=1/12
• EXCEL =(1/4)+(1/6)+(1/12)+(1/6)+(1/9)+(1/18)+(1/12)+(1/18)+(1/36)
Bernoulli Trials with 2 possible
outcomes.
• A Bernoulli trial is a random experiment in which
there are only two possible outcomes - success
and failure. If p is the probability of success, then
q=1-p is the probability of failure. Often we are
interested in the number of successes without
considering their order. The probability of exactly
k successes in n repeated trials is:
 n  k n-k
• b(k,n,p)=   p q
k 
Example: Trials with 2 possible
outcomes.
• A coin is tossed 6 times, H=success ,T=failure.
• n=6, p=q=1/2
• The probability of two heads, (k=2)
• Binomial coefficient
6
• b(2,6,1/2)=   (1/2)2(1/2)4 =15/64
2
Identically Distributed variable
Same probability distributions