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Circular Permutation
Continuation of Ch. 2
CIRCULAR PERMUTATIONS
The number of permutations of n distinct objects arranged in a
circle is (n 1)! where one of the objects is considered fixed in
its location.
Examples:1. In how many ways can a party of 4 (A,B,C, & D)
persons
arrange themselves around a circular table?
Answer: (4-1)! = 3! = 6
1. ABCD
A A---fixed
B
C
D
2. ABDC
3. ACDB
4. ACBD
5. ADBC
6. ADCB
In how many ways will A & B sit in adjacent seats? Ans.
4
CIRCULAR PERMUTATIONS
2. In how many ways can a party of 6 (A,B,C, D,E,& F)
persons arrange themselves around a circular table?
Answer: (6-1)! = 5! = 120 ways
In how many ways will:a.) A & B sit in adjacent seats?
(2)( 4! ) = 48
b.) A B & C sit in adjacent seats?
(3!)(3!)=36
CIRCULAR PERMUTATIONS
c. A B & C must not sit in adjacent seats?
d. the male (ACE) and the female (BDF) sit alternately?
1. A B C D E F 5. A F C D E B 9. A F E B C D2. A B E D C F 6. A
D C F E B 10. A F E D C B3. A B C F E D 7. A D C B E F 11. A D E F
C B4. A F C B E D 8. A B E F C D 12. A D E B C F
(1)(2!)(3!) = 12
5! 36 = 84
REMALYN QUINAY-CASEM
CHAPTER 3Conditional Probability and Independence
Conditional ProbabilityMultiplication Theorem for Conditional
ProbabilityPartition Rule and Bayes TheoremIndependence,
Independent or Repeated Trials
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Cond
ition
al P
roba
bilit
y Suppose that we toss 2 dice, and supposethat each of the 36
possible outcomes isequally likely to occur and hence
hasprobability 1/36. Suppose further that weobserve that the first
die is a 3. Then,given this information, what is theprobability
that the sum of the 2 diceequals 8?
Cond
ition
al P
roba
bilit
y
In conditional probability problems, the samplespace may be
reduced.
Conditional probability= # of outcomes that satisfy the
conditions /
# of outcomes in the sample space
Cond
ition
al P
roba
bilit
y
A die is rolled. Find the probability that a 3comes up if it is
known that an oddnumber turns up.
Let T be the event in which a 3 turns upand Q be the event in
which an oddnumber turns up.
Cond
ition
al P
roba
bilit
y
A coin is tossed; then a die is rolled. Findthe probability of
obtaining a 6, given thatheads comes up.
Let S be the event in which a 6 is rolled,and let H be the event
in which headscomes up.
Cond
ition
al P
roba
bilit
y
Two dice were thrown, and a friend tells usthat the numbers that
came up weredifferent. Find the probability that the sumof the two
numbers was 4.
Let D be the event in which the two diceshow different numbers,
and let F be theevent in which the sum is 4.
Cond
ition
al P
roba
bilit
y
Two dice are rolled, and a friend tells youthat the first die
shows a 6. Find theprobability that the sum of the numbersshowing
on the two dice is 7.
Let S1 be the event in which the first dieshows a 6, and let S2
be the event in whichthe sum is 7.
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Cond
ition
al P
roba
bilit
y
Assume that 2 cards are drawn in succession and without
replacement from a standard deck of 52 cards. Find the probability
thata. the second card is the ace of hearts, given that
the first card was the ace of spades.b. the second card is a
king, given that the first
card was a king.c. the second card is a 7, given that the first
card
was a 6.
Cond
ition
al P
roba
bilit
y
A coin is flipped twice. Assuming that allsample points are
equally likely, what is theprobability that both flips land on
heads,given that:(a) the first flip lands on heads?(b) at least one
flip lands on heads?
Cond
ition
al P
roba
bilit
y
Suppose that you hold a ticket in a lotterygame (1-30 numbers)
with the numbers 1,14, 15, 20, 23 and 27. You turn on
yourtelevision to watch the drawing but all yousee is one number,
15, being drawn whenthe power suddenly goes off in your house.You
dont even know whether 15 was thefirst, last, or some in-between
draw. What isthe probability that your ticket bears thewinning
number combination?
1 / 29C5 = 0.0000084
Cond
ition
al P
roba
bilit
y
In the card game bridge, the 52 cards aredealt out equally to 4
players called East,West, North, and South. If North andSouth have
a total of 8 spades amongthem, what is the probability that East
has3 of the remaining 5 spades?
(5C3 x 21C10) / 26C13 = 0.339
Cond
ition
al P
roba
bilit
y
The likelihood of a fatal vehicular crash is affected bynumerous
factors. The fatal crashes by speed limit andland use during 2004
are given in the table that follows.
Suppose a 2004 fatal crash is selected at random.What is the
probability that it occurreda. in a rural area?
Cond
ition
al P
roba
bilit
y
The likelihood of a fatal vehicular crash is affected bynumerous
factors. The fatal crashes by speed limit andland use during 2004
are given in the table that follows.
Suppose a 2004 fatal crash is selected at random.What is the
probability that it occurredb. in an area with a speed limit of no
more than
50 mph?
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Cond
ition
al P
roba
bilit
yThe likelihood of a fatal vehicular crash is affected
bynumerous factors. The fatal crashes by speed limit andland use
during 2004 are given in the table that follows.
Suppose a 2004 fatal crash is selected at random.What is the
probability that it occurredc. in a rural area, given that the
speed limit was no
more than 40 mph?
Cond
ition
al P
roba
bilit
y
The likelihood of a fatal vehicular crash is affected bynumerous
factors. The fatal crashes by speed limit andland use during 2004
are given in the table that follows.
Suppose a 2004 fatal crash is selected at random.What is the
probability that it occurredd. in an urban area, given that the
speed limit was
no more than 40 mph?
Cond
ition
al P
roba
bilit
y
EXERCISES
Cond
ition
al P
roba
bilit
y
EXERCISES
Cond
ition
al P
roba
bilit
y
8. Suppose that two dice were rolled and itwas observed that the
sum of the twonumbers was odd. Determine theprobability that the
sum was less than 8.
Cond
ition
al P
roba
bilit
y
9. The numbers of workers, in thousands, in the countryworkforce
in 2004 are shown in the table.
What is the probability that a randomly selected worker is
a.male who is at least 65 years of age?b. What is the probability
that a randomly selected worker is
a female?c. What is the probability that a randomly selected
worker
between 16 and 24 years old is a male?d. What is the probability
that a randomly selected female
worker is between 25 and 64 years of age?
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Mul
tiplic
atio
n Th
eore
m
P(AB) = P(B) P(AB)
Mul
tiplic
atio
n Th
eore
m
P(AB) = P(B) P(AB)
Suppose that an urn contains 8 red ballsand 4 white balls. We
draw 2 balls from theurn without replacement. What is
theprobability that both balls drawn are red?
P(R1 R2) = P(R1) P(R2 R1)
P(R1 R2) = (8/12) (7/11)P(R1 R2) = 0.4242
Mul
tiplic
atio
n Th
eore
m Generalized Multiplication RuleM
ultip
licat
ion
Theo
rem An ordinary deck of 52 playing cards is
randomly divided into 4 piles of 13 cardseach. Compute the
probability that eachpile has exactly 1 ace.
Mul
tiplic
atio
n Th
eore
m An ordinary deck of 52 playing cards israndomly divided into 4
piles of 13 cardseach. Compute the probability that eachpile has
exactly 1 ace.
Inde
pend
ence
, Ind
epen
dent
or
Repe
ated
Tria
ls
Suppose that youre rolling a fair six-sided die. IfA is the
event that the die comes up 1 and B isthe event that the die comes
up odd, are thesetwo events independent?
NO!
or P(B|A) = P (B)
The conditional probability of A given B equals theprobability
of A.
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Inde
pend
ence
, Ind
epen
dent
or
Repe
ated
Tria
ls
Two coins are tossed. Let E1 be the event the firstcoin comes up
tails, and let E2 be the event thesecond coin comes up heads. Are
E1 and E2independent?
We have two urns, I and II. Urn I contains 2 redand 3 black
balls, whereas urn II contains 3 redand 2 black balls. A ball is
drawn at random fromeach urn. What is the probability that both
ballsare black?
independent
independent (3/5) (2/5) = 6/25 Inde
pend
ence
, Ind
epen
dent
or
Repe
ated
Tria
ls
Suppose that two machines 1 and 2 in a factoryare operated
independently of each other.Machine 1 will become inoperative
during a given8-hour period with a probability of 1/3. Machine2
will become inoperative during the same periodwith a probability of
1/4. Determine theprobability that at least one of the machines
willbecome inoperative during the given period.
P(A or B)P(A or B) = P(A) + P(B) P(A and B)
P(A or B) = 1/3 + 1/4 (1/3)(1/4)
P(A or B) = 0.5
Inde
pend
ence
, Ind
epen
dent
or
Repe
ated
Tria
ls
or P(B|A) = P (B)
The conditional probability of A given B equals themarginal
probability of A.
Inde
pend
ence
, Ind
epen
dent
or
Repe
ated
Tria
lsBob is taking Math, Spanish, and English. Heestimates that his
probabilities of receiving As inthese courses are 1/10, 3/10, and
7/10,respectively. If he assumes that the grades can beregarded as
independent events, find theprobability that Bob makes
(a) all As (event A).
(b) no As (event N).
(c) exactly two As (event T).
Inde
pend
ence
, Ind
epen
dent
or
Repe
ated
Tria
ls
If two events are independent, does it mean thatthey cant happen
at the same time? NO!
If two events are independent, does it mean thatthey are always
mutually exclusive? NO!
Why? Cite an instance.
Why? Cite an instance.
Part
ition
Rul
e an
d Ba
yes
The
orem
E = EF + EFC
PARTITION RULE/ Total Probability
P(E )= P(EF) + P(EFC )
P(E )= P(EF)P(F) + P(EFC ) P(FC ) ORP(E )= P(EF)P(F) + P(EFC )
(1 - P(F))
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Cond
ition
al P
roba
bilit
y an
d PA
RTIT
ION
S
(60/100) (1/2) + (10/20) (1/2) = 0.46667
Two boxes contain long bolts and shortbolts. Suppose that one
box contains 60long bolts and 40 short bolts, and that theother box
contains 10 long bolts and 20short bolts. Suppose that one box
isselected at random and a bolt is thenselected at random from that
box, what isthe probability that this bolt is long?
PARTITION RULE/ Total ProbabilityP(E )= P(EF)P(F) + P(EFC ) P(FC
)E
Part
ition
Rul
e an
d Ba
yes
The
orem
PARTITION RULE/ Total ProbabilityP(E )= P(EF)P(F) + P(EFC ) P(FC
)E
An insurance company believes that people can be divided intotwo
classes: those who are accident prone and those who arenot. The
companys statistics show that an accident-proneperson will have an
accident at some time within a fixed 1-yearperiod with probability
.4, whereas this probability decreases to.2 for a person who is not
accident prone. If we assume that 30percent of the population is
accident prone, what is theprobability that a new policyholder will
have an accident withina year of purchasing a policy?
(0.4)(0.3) + (0.2)(0.7) = 0.26Suppose that a new policyholder
has an accident within a yearof purchasing a policy, what is the
probability that he or she isaccident prone? (0.3)(0.4) / 0.26 =
0.4615
Part
ition
Rul
e an
d Ba
yes
The
orem
EXAMPLESuppose that customers have three restaurants to
choosefrom in a certain town: R1, R2, R3. Previous data
collectionhas shown that these restaurants get 50%, 30% and 20%
ofthe customers, respectively. Suppose you also know that 70%of the
customers who dine at R1 are satisfied (and 30% arenot), 60% of
those who dine at R2 are satisfied, and 50% ofthe customers who
dine at R3 are satisfied. What is theprobability that someone who
eats at a restaurant in thistown will be satisfied?
0.5
0.3
0.2
0.70.3
0.60.4
0.50.5
0.5 0.7 =Joint ProbabilitiesFirst R second S
0.150.180.120.10.1
0.35
P(S) = 0.35 + 0.18 + 0.1 = 0.63
P(E )= P(EF)P(F) + P(EFC ) P(FC )
Part
ition
Rul
e an
d Ba
yes
The
orem
PARTITION RULE/ Total ProbabilityP(E )= P(EF)P(F) + P(EFC ) P(FC
)E
A company buys microchips from three suppliersI, II, andIII.
Supplier I has a record of providing microchips that contain10%
defectives; Supplier II has a defective rate of 5%; andSupplier III
has a defective rate of 2%. Suppose 20%, 35%, and45% of the current
supply came from Suppliers I, II, and III,respectively. If a
microchip is selected at random from thissupply, what is the
probability that it is defective?
0.20(0.10) + 0.35(0.05) + 0.45(0.02) = 0.0465
Part
ition
Rul
e an
d Ba
yes
The
orem
Bayes Formula
Part
ition
Rul
e an
d Ba
yes
The
orem
Bayes Formula
Based from the partition ruleP(E )= P(EF)P(F) + P(EFC ) P(FC
)
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Part
ition
Rul
e an
d Ba
yes
The
orem
A company buys microchips from three suppliersI, II, andIII.
Supplier I has a record of providing microchips that contain10%
defectives; Supplier II has a defective rate of 5%; andSupplier III
has a defective rate of 2%. Suppose 20%, 35%, and45% of the current
supply came from Suppliers I, II, and III,respectively. If a
microchip is selected at random from thissupply, what is the
probability that it is defective?
0.20(0.10) + 0.35(0.05) + 0.45(0.02) = 0.0465
Bayes Formula
P(A|B) = P(B|A) P(A)P(B|A) P(A) + P(B|AC) P(AC )
If a randomly selected microchip is defective, what is
theprobability that it came from Supplier II?
(0.05) (0.35) / 0.0465 = 0.376
Part
ition
Rul
e an
d Ba
yes
The
orem
EXAMPLESuppose that customers have three restaurants to
choosefrom in a certain town: R1, R2, R3. Previous data
collectionhas shown that these restaurants get 50%, 30% and 20%
ofthe customers, respectively. Suppose you also know that 70%of the
customers who dine at R1 are satisfied (and 30% arenot), 60% of
those who dine at R2 are satisfied, and 50% ofthe customers who
dine at R3 are satisfied. What is theprobability that someone who
eats at a restaurant in thistown will be satisfied?
0.5
0.3
0.2
0.70.3
0.60.4
0.50.5
0.5 0.7 =Joint ProbabilitiesFirst R second S
0.150.180.120.10.1
0.35
P(S) = 0.35 + 0.18 + 0.1 = 0.63
Part
ition
Rul
e an
d Ba
yes
The
orem
Bayes Formula
P(A|B) = P(B|A) P(A)P(B|A) P(A) + P(B|AC) P(AC )
EXAMPLEWhats the chance that a customer ate at R2,given that he
or she is satisfied?
0.5
0.3
0.2
0.70.3
0.60.4
0.50.5
0.5 0.7 =Joint ProbabilitiesFirst R second S
0.150.180.120.10.1
0.35
P(S) = 0.35 + 0.18 + 0.1 = 0.63
P(R2|S)= 0.18/0.63= 0.286
or 28.6%
Part
ition
Rul
e an
d Ba
yes
The
orem
Bayes Formula
P(A|B) = P(B|A) P(A)P(B|A) P(A) + P(B|AC) P(AC )
EXAMPLEAssuming that the costumer is satisfied, whichrestaurant
was he or she most likely to have eaten at?
0.5
0.3
0.2
0.70.3
0.60.4
0.50.5
0.5 0.7 =Joint ProbabilitiesFirst R second S
0.150.180.120.10.1
0.35
P(S) = 0.35 + 0.18 + 0.1 = 0.63P(R2|S) = 0.286
P(R1|S)= 0.35/0.63= 0.556
or 55.6%
Part
ition
Rul
e an
d Ba
yes
The
orem
Bayes Formula
P(A|B) = P(B|A) P(A)P(B|A) P(A) + P(B|AC) P(AC )
EXAMPLEAssuming that the costumer is satisfied, whichrestaurant
was he or she most likely to have eaten at?
0.5
0.3
0.2
0.70.3
0.60.4
0.50.5
0.5 0.7 =Joint ProbabilitiesFirst R second S
0.150.180.120.10.1
0.35
P(S) = 0.35 + 0.18 + 0.1 = 0.63P(R2|S) = 0.286
P(R3|S)= 0.1/0.63= 0.159
or 15.9%
P(R1|S) = 0.556at Restaurant 1
Part
ition
Rul
e an
d Ba
yes
The
orem
This makes sense because Restaurant 1 hasthe most customers and
gets the highestcostumer satisfaction rating.
EXAMPLEAssuming that the costumer is satisfied, whichrestaurant
was he or she most likely to have eaten at?
0.5
0.3
0.2
0.70.3
0.60.4
0.50.5
0.5 0.7 =Joint ProbabilitiesFirst R second S
0.150.180.120.10.1
0.35
P(S) = 0.35 + 0.18 + 0.1 = 0.63P(R2|S) = 0.286
P(R3|S)= 0.1/0.63= 0.159
or 15.9%
P(R1|S) = 0.556at Restaurant 1
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Part
ition
Rul
e an
d Ba
yes
The
orem
An industrial company manufactures a certaintype of car in three
towns called Farad,Gilbert, and Henry. Of 1000 made in Farad,20%
are defective; of 2000 made in Gilbert,10% are defective, and of
3000 made in Henry,5% are defective. You buy a car from a
distantdealer. Let D be the event that it is defective, Fthe event
that it was made in Farad and so on.Find: (a) P(F|Hc); (b)
P(D|Hc);
(c) P(D); (d) P(F|D).Assume that you are equally likely to have
bought any one of the 6000 cars produced. P
artit
ion
Rule
and
Ba
yes
The
orem
(a) P(F|Hc) conditional probability
Part
ition
Rul
e an
d Ba
yes
The
orem
(b) P(D|Hc) conditional probability
Hc = F G
60003000
6000200
+6000200
=
=3000400
=152
F G =
What if F G =
Part
ition
Rul
e an
d Ba
yes
The
orem
(c) P(D) total probability/partition rule
60003000
152
+60003000
201
=
151
+401
=
12011
=
Part
ition
Rul
e an
d Ba
yes
The
orem
(d) P(F|D) conditional probability
1201160001000
51
=
=
12011301
=114 Pa
rtiti
on R
ule
and
Baye
s T
heor
em
Suppose there is a school having 60% boysand 40% girls as
students. The femalestudents wear trousers or skirts in
equalnumbers; the boys all wear trousers. Anobserver sees a
(random) student from adistance; all the observer can see is that
thisstudent is wearing trousers. What is theprobability this
student is a girl?
first stage = sexsecond stage = uniform
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Part
ition
Rul
e an
d Ba
yes
The
orem
A laboratory blood test is 95 percent effective indetecting a
certain disease when it is, in fact,present. However, the test also
yields a falsepositive result for 1 percent of the healthypersons
tested. (That is, if a healthy person istested, then, with
probability .01, the test resultwill imply that he or she has the
disease.) If .5percent of the population actually has the
disease,what is the probability that a person has thedisease given
that the test result is positive?
1st= real health condition2nd = test result Pa
rtiti
on R
ule
and
Baye
s T
heor
em
Consider two urns. The first contains two whiteand seven black
balls, and the second containsfive white and six black balls. We
flip a fair coinand then draw a ball from the first urn if the
coinlands on heads. If tails comes up, draw a ballfrom the second
urn. What is the probability thatthe outcome of the toss was heads
given that awhite ball was selected?Let W be the event that a white
ball is drawn, andlet H be the event that the coin comes up
heads.
))P(HH|P(W + H)P(H)|P(WH)P(H)|P(W
=W)|H( ccP
21
115
+ 21
92
21
92
= =
1986791
= 67
22first stage = ?second stage = ?
Cond
ition
al P
roba
bilit
y an
d In
depe
nden
ce
EXERCISESCo
nditi
onal
Pro
babi
lity
and
Inde
pend
ence
EXERCISES
Cond
ition
al P
roba
bilit
y an
d In
depe
nden
ce
EXERCISES
Cond
ition
al P
roba
bilit
y an
d In
depe
nden
ce
EXERCISES
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Cond
ition
al P
roba
bilit
y an
d In
depe
nden
ce
EXERCISES
Cond
ition
al P
roba
bilit
y an
d In
depe
nden
ce
EXERCISES
11. A jar contains 7 red, 6 green, 8 blue, and 4yellow marbles.
A marble is chosen atrandom from the jar. After replacing it,
asecond marble is chosen. What is theprobability of choosing
a. a red and then a yellow marble?b. 2 yellow marbles?c. no blue
marbles?d. 2 marbles of the same color?e. at least one red
marble?
Cond
ition
al P
roba
bilit
y an
d In
depe
nden
ce
EXERCISES
12. The probability that an archer hits thetarget when it is
windy is 0.4; when it is notwindy, her probability of hitting the
targetis 0.7. On any shot, the probability of agust of wind is 0.3.
Find the probabilitythat:
(a) On a given shot, there is a gust of windand she hits the
target.
(b) She hits the target with her first shot.
(c) She hits the target exactly once in two shots. Con
ditio
nal P
roba
bilit
y an
d In
depe
nden
ce
EXERCISES
13. You enter a chess tournament where yourprobability of
winning a game is 0.3 againsthalf the players (call them type 1),
0.4against a quarter of the players (type 2),and 0.5 against the
remaining quarter ofthe players (type 3). You play a gameagainst a
randomly chosen opponent. Whatis your probability of winning?
Cond
ition
al P
roba
bilit
y an
d In
depe
nden
ce
EXERCISES
14. You have a blood test for some rare diseasethat occurs by
chance in 1 in every 100 000people. The test is fairly reliable; if
youhave the disease, it will correctly say sowith probability 0.95;
if you do not havethe disease, the test will wrongly say you dowith
probability 0.005. If the test says youdo have the disease, what is
the probabilitythat this is a correct diagnosis?
Cond
ition
al P
roba
bilit
y an
d In
depe
nden
ce
EXERCISES
15. Two methods, A and B, are available forteaching a certain
industrial skill. Thefailure rate is 30% for method A and 10%for
method B. Method B is moreexpensive, however, and hence is used
only20% of the time. (Method A is used theother 80% of the time.) A
worker is taughtthe skill by one of the two methods, but hefails to
learn it correctly. What is theprobability that he was taught by
usingmethod A?
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Cond
ition
al P
roba
bilit
y an
d In
depe
nden
ce
EXERCISES
16. John flies frequently and likes to upgradehis seat to first
class. He has determinedthat if he checks in for his flight at
least 2hours early, the probability that he will getthe upgrade is
0.8; otherwise, theprobability that he will get the upgrade is0.3.
With his busy schedule, he checks in atleast 2 hours before his
flight only 40% ofthe time. What is the probability that for
arandomly selected trip John will be able toupgrade to first
class?
End of Presentation