Session 2 2014 Version Part 1b

1/10/2015

1

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


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


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

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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


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


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

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bilit

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EXERCISES

Cond

ition

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EXERCISES

Cond

ition

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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

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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

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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


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


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


0.150.180.120.10.1

0.35

P(S) = 0.35 + 0.18 + 0.1 = 0.63

Part

ition

Rul

e an

d Ba

yes

The

orem

Bayes Formula


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


0.150.180.120.10.1

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


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


0.150.180.120.10.1

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



0.5

0.3

0.2

0.70.3

0.60.4

0.50.5


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.


0.5

0.3

0.2

0.70.3

0.60.4

0.50.5


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

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EXERCISESCo

nditi

onal

Pro

babi

lity

and

Inde

pend

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EXERCISES

Cond

ition

al P

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depe

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EXERCISES

Cond

ition

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EXERCISES

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EXERCISES

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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

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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

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d In

depe

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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

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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

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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

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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

Documents

Session 2 2014 Version Part 1b