Classical Probability
Example 4.3: Flip a coin three times, what is the probability of
(a) There are 2 Heads only?
(b) There are Heads only?
(c) There are 2 Heads, given the first is Head?
(d) There are 2 Heads, given the first is Tail?
Example 4.4 There is a box contains 6 white balls and 4 black
balls. Random pick up 2 balls, calculate the probability of
event that both balls are white, in 3 cases:
(a) Pick up one, replace, then the next
(b) Pick up one by one, without replacement
(c) Pick up 2 balls simultaneously

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Lecture 4. BASIC PROBABILITY
Probability
Outcome – Event
Complement Event, Intersection Event, Union Event
Mutually Exclusive, Independent, Collectively
Exhausive, Partitions
Bernoulli Formula
Total Probability, Bayes’ Theorem
[1] Chapter 4, pp. 169 - 212
PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 1
“Problem of Points”
2 players A and B, contributed 50 Franc each.
The game is winner – loser only (no draw), A and B are
equally-likely to win in each match.
Game rule: play 9 matches, the one wins more is final
winner and takes all of 100F
But the game had stopped after 7 matches, and scores
at that time of A and B are 4 and 3, respectively.
How could they distribute the money?
PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 2
4.1. Probability
Probability is quantitative measure of uncertainty, the
chance that an uncertain event will occur.
Probability: Subjective and Objective
Probability of event A is denoted by P(A)
0 ≤ P(event) ≤ 1
P(always) = 1
P(impossible) = 0
() > (): A is more possible to occur than B
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Experiment – Outcome - Event
Experiment Outcome Event
Flip a coin Head, tail ‘Head’, ‘tail’
Toss a die 1,2,3,4,5,6 dot(s) ‘greater than 3’
Do an exam Score = 0, 1, 2,, 10 ‘pass’;
‘excellent’
Invest in a project Profit: (+), (–), zero ‘Non negative’
‘profitable’
Apply for a job Pass, fail
Do a job Salary =
PROBABILITY & STATISTICS– Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 4
Classical Probability
Sample space: all basic outcomes, denoted by Ω
Assumes all basic outcomes in the sample space are
equally-likely to occur
Number of basic outcomes:
Number of basic outcomes for event A:
Probability of event A:
=
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Classical Probability
Example 4.1. Flipping a “fair” coin once, the probability
that the Head is up = ?
Example 4.2. Flipping a fair coin two times. What is the
probability of
(a) “There are 2 Heads”
(b) “There are 1 Head, 1 Tail”
(c) “There are 2 Tails”
PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 6
Classical Probability
Example 4.3: Flip a coin three times, what is the probability of
(a) There are 2 Heads only?
(b) There are Heads only?
(c) There are 2 Heads, given the first is Head?
(d) There are 2 Heads, given the first is Tail?
Example 4.4 There is a box contains 6 white balls and 4 black
balls. Random pick up 2 balls, calculate the probability of
event that both balls are white, in 3 cases:
(a) Pick up one, replace, then the next
(b) Pick up one by one, without replacement
(c) Pick up 2 balls simultaneously
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Example
Male Female Sum
Freq. 160 240 400
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Male Female Sum
Prob. 0.4 0.6 1
Freq. Male Female Sum
Single 60 80 140
Married 100 160 260
Sum 160 240 400
Prob. Male Female Sum
Single 0.15 0.2 0.35
Married 0.25 0.4 0.65
Sum 0.4 0.6 1
Frequency Table Probability Table
Cross-Frequency Table Joint Probability Table
4.2. Probability vs Proportion
Number of experiments:
Frequency of event A:
Proportion of A:
When n is large enough: () ≈ /
Ex. In 100,000 new-borns, there were 51,000 boys, then
the probability of “new-born is boy” is about 0.51
Ex. In 4,000 students, there are 600 fail in subject A, then
probability of “Fail in subject A” is about 0.15
PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 9
4.3. Complement Event
Complement of A : all outcomes that not belong to A
Denoted by Ā
Ex.
A = “two flipped coins are Heads” A = ?
B = “both picked balls are White” B = ?
C = “all of students passed” C = ?
PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 10
A
Ω
Ā
Law of Complement
Law: () = – ()
Ex. Flipping coin twice, what is the Probability of “Have at
least one Tail” ?
Complement of “Have at least one tail” is “No any
Tail”, or “Two Heads”
Probability = 1 – P(Two Heads) = 1 – ¼ = ¾
PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 11
4.4. Intersection Event
Intersection of A and B: all outcomes that belong to
both A and B,
Denoted by ∩
∩ ̅ = Ø
Ex. Pick 2 balls from the box of Blacks and Whites, A is
“The first is white” , B is “The second is white”
AÇB is “Both balls are white”
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A B
Ω
AÇB
Mutually Exclusive
A and B are Mutually exclusive : have no common
i.e., A Ç B = Ø
A and Ā ?
Ex. Pick up 2 balls, A = “two Whites”, B = “two Blacks”, C =
“At least one White”
A and B are mutually exclusive; B and C are mutually
exclusive, but A and C are NOT mutually exclusive,
PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 13
A B
Ω
Conditional Probability
Conditional probability : probability of one event,
given that another event has occurred.
The conditional probability of B given that A has
occurred, (B given A) denoted by P(B | A)
Ex. Flip a coin 3 times
P(2 Heads) = 3/8 = 0.375
P(2 Heads | The first is Head) = 2/4 = 0.5
P(2 Heads | The first is Tail) = 1/4 = 0.25
P(2 Heads | There are at least one Head) =
PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 14
Conditional Probability
Ex. Pick up one ball from box of 6 Whites and 4 Blacks,
then one more. Let A = “the first ball is White”, B =
“The second ball is White”
Without replacement of the first ball: P(B | A) = 5/9
Replacement of the first ball: P(B | A) = 6/10
Ex. Pick up 3 balls one-by-one, without replacement
P(The third is White | Two firsts are Whites) =
P(The third is White | Two firsts are 1 White 1 Black) =
P(The third is White | Two firsts are Blacks) =
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Independent Events
A and B are independent : A does not affect B, and B does
not affect A
Û ( | ) = () and (|) = ()
A and B are not independent dependent
Ex. From box of 6 Whites, 4 Blacks, pick up one by one
Let A = “the 1st is White”, B = “The 2nd is White”.
Replace the first A and B are independent
(|) = (|Ā) = 6/10 ; (|) = 6/10
Without replacement A and B are dependent
(|) = 5/9 ; (|Ā) = 6/9
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Law of Intersection
∩ = × = × (|)
A and B are independent
Û ∩ = × ()
Conditional Probability
=
∩
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Law of Intersection
Ex. Pick one ball from box of 6 whites and 4 blacks, then
one more. Let A = “the first ball is White”, B = “The
second ball is White”.
The Probability of “Two White” = (Ç)
(Ç) = () × (|)
Without replacement
Ç = × =
Replacement
Ç = × =
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Example
Example 4.5. The chance that a student passes the
subjects A and B are 0.6 and 0.8, respectively.
However, if passed subject A, the chance for him to
pass subject B is 0.9
(a) What is probability that he passes both subjects?
(b) Whether “Pass subject A” and “Pass subject B” are
independent?
(c) What is probability of passing subject A, given B has
been passed?
PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 19
In General
Intersection of n events A1 and A2 and and An
∩ ∩ ⋯ ∩
A1, A2,,An are totally independent Û
( ∩ ∩ ⋯ ∩ ) = () × ⋯ × ()
Ex. Pick 5 balls from box of 6 Whites and 4 Backs, with
replacement, the Probability that all of them are white is
6
10
×
6
10
×
6
10
×
6
10
×
6
10
= 0.6
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4.5. Union Events
Union of A and B: all outcomes belong to either A or B
Denoted by ∪
∪ includes:
∩ ; ∩ ; ̅ ∩
Ex. Pick 2 balls from box of 6Ws, 4Bs
A = “the first is W”, B = “the second is W”,
then AB is “at least one W”
PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 21
A B
Ω
Law of Union
∪ = + − ( ∩ )
A and B are mutually exclusive events
Û ∩ = Ø Û ( ∩ ) = 0
Û ( ∪ ) = () + ()
PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 22
A B
Ω
A B
Ω
Example
Example 4.6. The chance that a candidate passes the
subjects A and B are 0.6 and 0.8, respectively.
However, if passed subject A, the chance for him to
pass subject B is 0.9. What is probability that (s)he:
(a) pass both subjects?
(b) pass at least one subject?
(c) fail in both subjects?
(d) fail in at least one subject?
PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 23
4.6. Joint Probability
Ex. The chance that a candidate passes the subjects A
and B are 0.6 and 0.8, respectively. The chance of passing
both is 0.54. Building the join probability table
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Join
Probability
Table
Subject B
Pass
B
Fail
B
Sum
Sub.
A
Pass A 0.54 0.06 0.6
Fail A 0.26 0.14 0.4
Sum 0.8 0.2 1
Joint Probability
Example 4.7. Base on the Join Probability table, build the
Marginal probability distribution, and what is the
probability of
(a) Pass B given passed A?
(b) Pass B given failed A?
(c) Fail A given passed B?
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Join
Probability
Table
Subject B
Pass
B
Fail
B
Sum
Sub.
A
Pass A 0.54 0.06 0.6
Fail A 0.26 0.14 0.4
Sum 0.8 0.2 1
Marginal prob. distributions
A A
P 0.6 0.4
B B
P 0.8 0.2
Example
Example 4.8. The exam includes two independent
questions. The probability that student’s answer is
correct in the A and B question are 0.7 and 0.5. What
is the probability that a student
(a) Correct in at least one answer?
(b) Correct in only one answer?
(c) Incorrect in at least one answer?
(d) Incorrect in both answers?
(e) Joint probability table?
PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 26
Problem
Problem 4.9. Base on the table, fill the blanks, and build
the marginal prob. tables; what is the proportion of:
(a) the customer who use services
(b) good only buyers in female customers
(c) services users only in male customers
(d) male in customers who buy goods only
(e) female in customers who use services
PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 27
Join prob.
of
Customers
Buy goods
only
(BO)
Use Services
only
(SO)
Buy goods and
use services
(BS)
Sum
Male: (M) 0.2 0.1 0.15 0.45
Female: (F) 0.15 0.1 0.3 0.55
Sum 0.35 0.2 0.45 1
In General
Union of A1 or A2 or or An is denoted by
A1 A2 An
A1, A2,,An are totally mutually exclusive
P(A1A2An) = P(A1) + P(A2) ++ P(An)
Example 4.10. There are 3 optional subjects 1, 2, 3. Each of two
students B and C randomly chooses one subject, assumed that they
are independently. What the probability that they choose the same
subject?
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Collectively Exhaustive & Partitions
A1, A2,..., An are Collectively Exhaustive:
A1A2 ... An = Ω
A1,A2,...,An are Collectively Exhaustive and Mutually
Exclusively
They are Partitions
A and Ā are Partitions
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A1 A2 A3 ... An
Ω
A1, A2,..., An are Partitons : P(A1) + P(A2) ++ P(An) = 1
Ex. Flip coin twice; A1 is (2 Heads), A2 is (2 Tails), A3 is (1
Head 1Tail), A4 is (At least one Head).
Whether the following groups are Collectively
Exhaustive, Partitions, Complement?
(a) A1,A2,A3
(b) A1,A2,A4
(c) A1,A3,A4
(d) A2, A4
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4.7. Bernoulli Formula
Problem 4.11. An quiz has 3 independent questions, the
chance for a candidate to answer correctly each
question are equally, and be 0.6. What is the
probability that a candidate will be
(a) Correct in only one answer?
(b) Correct in only two answers?
(c) Correct in at least one answer?
PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 31
Bernoulli Formula
There are independent experiments
The probability that event A occurs in each
experiment is equally and denoted by , then
probability of A do not occur is – .
Bernoulli Trial, denoted by B(n, p)
The probability that in n experiments, event A occurs
in times:
, =
−
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Example
Example 4.12. The test includes 10 independent multiple
choice questions. Each question has 4 options with only
one correct choice.
A candidate randomly answers all question. What is
the probability that (s)he:
(a) Correct in 4 answers
(b) Correct in at least one answer
PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 33
3.8. Total Probability – Bayes’ Theorem
Example 4.13. There are 2 boxes, Box type I contains 6
white balls and 4 black ones, Box type II contains 8
white balls and 2 black ones.
(a) Random choose one box, then random choose one
ball. What is the probability that the ball is white?
(b) If the chosen ball is white, what is the probability that
the chosen box is Type I, Type II?
(c) If the chosen ball is black, what is the probability that
the chosen box is Type I, Type II ?
PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 34
Problem
(White ball) =
= (Box I and White ball) or (Box II and White ball)
= (B1ÇW) (B2 ÇW)
=
×
+
×
= 0.3 + 0.4 = 0.7
Probability of 0.7 is contributed by 0.3 from Box I and
0.4 from Box II. Therefore Probability that given Ball is
white, chosen box is Box type I =
.
.
PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 35
Example
Use table to calculate and analyze Total probability
and contribution probability
Example: What happens if there are 3 boxes of type I,
and 2 boxes of type II ?
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() (|) (Ç) (|)
Box I (B1) 0.5 0.6 0.3 0.3 / 0.7
Box II (B2) 0.5 0.8 0.4 0.4 / 0.7
Sum 1 0.7 1
Bayes’ Theorem
B1,B2,...,Bk are partitions
Total probability:
() = ()(|) + ⋯ + ()(|)
= ∑ ()(|)
Bayes’ Theorem: Probability of Bi occurs, given A
occurred is P(Bi|A)
=
(|)
()
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Bayes’ Theorem
() : prior-probability
(|) : posterior-probability
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1st Stage
Prior: P(Bi)
2nd Stage
P(A|Bi) P(AÇBi)
Posterior
P(Bi|A)
B1 () ( |) ( ∩ ) ( ∩ ) / ()
B2 () ( |) ( ∩ ) ( ∩ ) / ()
... ... ... ... ...
Bk () ( |) ( ∩ ) ( ∩ ) / ()
Sum 1 P(A) 1
Key concepts
Events, Probability, Conditional probability
Complement, Partitions, Intersection, Union
Independent, Mutually exclusive
Probability Rules, Bernoulli formula
Bayes’ Theorem
PROBABILITY & STATISTICS – Bui Duong Hai – NEU – www.mfe.edu.vn/buiduonghai 39
Exercices
[1] Chapter 4
(p180) 11, 13
(p184) 18, 19
(p190) 25, 28,
(197) 33, 35,
(204) 41, 42
(208) 46, 50, 52, 57
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