Applicable Mathematics

FOUNDATION IN ARTS

PMTH 003

PROBABILITY

ASSIGNMENT20091.A class of 32 students with 10 boys decides to elect a class executive council which will oversee the daily running of the class and the issuing of duties. It is decided that the executive should consist of 7 students and that the selection process to be random.

(a) How many different executive councils can be selected from the class if there is no restriction placed on the gender of the council members?

(b) How many different executive councils can be selected if two of the classmates are on the student council and it is decided that they cannot be both on the executive committee?

2.The Returning Officer for an election of the Allied workers Union has summerised the positions vacant and nominations from each if the three member group of the union.

Management

Apprentices

Officer

Position vacant 6

2

3

Nominations

19

28

5

From the table it can be seen that 28 apprentices nominated for the two positions available to apprentices on the 11 person committee.

(a) How many different committees can be selected?

(b) Two of the apprentices are sisters. How many different ways can the committee of 11 be elected if the two sisters be on the committee together?

3.When a train stops at a suburban railway station, 12 passengers get off. Of these 12 passengers, 5 are males. There is only one taxi, which can only take 4 people, waiting at the station. Ignoring the seating order when in the taxi, find

(i) how many different selection of 4 people can enter the taxi?

(ii) How many selection of 2 males and 2 females can enter the taxi?

Of the 12 people, 3 are a family (2 males and 1 female), there are 2 females couples and the rest are people on their own. The family cannot be separated and neither can each couple.

(iii) how many different selection of 4 people can now enter the taxi?

(iv) When passenger are in the taxi, in how many different ways can they be arranged?

4.A test for detecting cancer which appears promising has been developed. It was found that 98% of the cancer patients in a large hospital react positively to the test, where as only 4% of those not having cancer did so. If 3% of patients in the hospital have cancer, what is the probability that a patient picked at random will react to the test. What is the probability that a patient picked at random who react positively will have cancer?

5.A group of 100 musicians are attending a party. Between them they can play four instruments.

None of the twenty drummers can play any other instruments

Ten musicians can each play 3 instruments.

Half of the non-drummers can play the violin.

Sixty of the musicians can play the guitar and twenty can play the flute.

Ten of the flautist can play the violin

Seventy of the musicians can play only one instrumentTwenty five guitarists can play the violin

Display this information in a Venn Diagram, indicating on your diagram how many musicians fall into each category.

6.A sporting club has 12 committee members of whom 6 are juniors and 6 are seniors. A subcommittee of 5 is to be chosen to attend a conference later in the year. Find the number of different sub-committee which contain

(a) 2 juniors & 3 seniors

(b) 4 junior and 1 senior

(c) At most 2 juniors

(d) a majority of juniors if there must be at least one senior on the committee

7.The 90 passengers on a luxury cruise were questions as to which of tennis, squash and badminton they played. It was found that 28 played badminton, 67 played tennis, 27 played squash, 2 played all three, 8 played badminton and tennis but not squash, and all squash players played tennis.

How many played:

(a) none of the three given sports?

(b) exactly one of the three given sports?(c) exactly two of the three given sports?

(d) squash, but not badminton?

8.How many ways can 9 people (5 males and 4 females) be seated in a row if

(i) there is no restriction?

(ii) there must be one male between two females?

(iii) two of the females (Alice and Betty), refuse to sit next to each other?

(iv) Alice and Betty will be seated together?

9.A telephone has ten buttons numbered from 0 to 9 inclusive. Assume repetition of numbers is possible.

(a) If a telephone number has 7 digits how many telephone numbers are possible?

From all 7 digits numbers how many

(b) begin with 0?

(c) begin with 0 and end with 5?

(d) are odd numbers?

(e) have the number 8 at either the front or the back but not both?

(f) have the number 9 at either front or back or both?

(g) begin with either a 7 or 9?

(h) have the number 5 only once?

10.In a television ratings survey involving four television stations, a polling organization decides to interview 200 persons. They are classified in the following table according to whether they were reached on the first, second or a later call of the interviewer and according to their preference for television station A, B, C or D.

Prefers T.V.

station

Interviewed onABCD

1st call

2nd call

3rd or later call24

7

238

18

358

20

420

5

1

Use the appropriate relative frequencies to

(i)find the conditional probability with which station C was preferred among those reached on the first call;

(ii)find the probability with which station B was preferred among all those interviewed;

(iii)find the probability with which an interviewee was reached on the first call among those preferring station A;

(iv)find the probability with which three or more calls were required among all those interviewed.

11.A and B are two events such that P(A) = .4 and P(AB) = .7. Find the value of P(B) if

(a) A and B are mutually exclusive.

(b) A and B are independent.

What is the value of P(A/B) in each case?

12.A box contains 2 Blue, 3 Red and one White ball. A ball is drawn from the box and then a second ball is drawn without replacement.

(a)List the sample space of this experiment.

(b)Find the following probabilities:

(i)both balls are red,

(ii) the balls are of a different colour,

(iii) the balls are not both white,

(iv) red ball is not drawn.

13.Statistics for two drugs Aman 1 and Aman 2 are gathered. It is found that 3% of people have and adverse reaction to Aman 1, 95% of people do not have an adverse reaction to Aman 2 while 0.2% of people react adversely to both Aman 1 and Aman 2.

(a)What percentage of people can take either drug without fear of an adverse reaction?

(b)Of those people who have and adverse reaction to Aman 1, what percentage of have an adverse reaction to Aman 2?

(c)If a person is chosen at random, what is the probability that the person will not react adversely to Aman 1 given that they reacted favourably to Aman 2?

14.A company is investigating the relationship between physical fitness and efficiency in the workplace. They found that 20% of the workers are fit. If a worker is fit there is a 73% chance their efficiency will increase over time. If a worker is unfit there is a 22% chance their efficiency will increase over time. Determine the probability a worker

(a)is fit and the workers efficiency increased over time.

(b)is not fit and workers efficiency increased over time,

(c)increased efficiency over time,

If a worker is known to improve efficiency over time, determine the probability the worker is

(d)fit

(e)unfit.

15.A subcontractor is known to two firms A and B. Both of these firms have submitted a tender for a major contract. One of these firms will get the contract. The probability that firm A gets the contract is 0.56. A firm A gets the contract the subcontractor has a 35% chance of obtaining work. If firm B gets the contract, there is a 56% chance the subcontractor will obtain work. Determine the probability that

(a)firm B gets the contract.(b)the subcontractor obtains work.

16.Space shuttles use O-rings to contain the burning fuel in the combustion chamber. Space shuttle Avenger uses three O-rings, O1, O2 and O3. For a successful launch there should be no fuel leakage and for that O1 has to function and at least one of O2 and O3 must function. Assuming that the O-rings behave independently of one another and that each O-ring has a 0.9 probability of working, calculate the following probabilities.

(i)there is a fuel leakage.

(ii)O2 functions and the launch is successful.

(iii)launching is successful given that O3 fails.

(iv)O3 fails given that the launch is successful.

17.100 students were each asked if they had watched TV, listened to the radio or played a tape the previous evening. 65 students had watched TV, 70 had listened to the radio, 40 had played a tape and 20 had done all three. 25 had watched TV and listened to the radio but not played a tape. 10 had listened to the radio and played a tape but not watched TV. 5 had only played a tape.

(i)Illustrate these results in a Venn diagram.

For this group of students calculate the probability that a randomly chosen student had

(ii)not listened to the radio,

(iii)not listened to the radio nor watched TV,

(iv)watched TV and listened to the radio,

(v)listened to the radio or had played a tape,

(vi)played a tape given that they had listened to the radio.

18.On the following Venn diagrams, shade the indicated regions.

(i)

(ii)

(iii)

19.In a group of 83 Year 12 students, 49 study Applicable Mathematics, 47 study Physics and 36 study Chemistry. Thirteen students study Physics and Chemistry but not Applicable Mathematics. Twenty students study Applicable Mathematics and Chemistry, whilst 26 study Applicable Mathematics and Physics. Exactly two out of the three subjects are studied by 41 students.

(a)Draw a Venn diagram to represent the above information, indicating clearly on your diagram how many students are in each category.

(b)For a student chosen from these Year 12 students, calculate the probability that the student studies

(i)all three subjects,

(ii) none of the three subjects,

(iii) Physics, given that the student studies Applicable Mathematics,

(iv) Applicable Mathematics, given that the student studies both Physics and Chemistry.

20.Shade the indicated areas on the following Venn diagrams.

(i)R ( (S ( T)

(ii)

(iii)(R ( S) ( (S ( T)

21.In a company with 141 staff, 120 of the staff have a desktop computer on their office desk, 40 have a laptop computer and 33 have a desktop computer at home provided by the company. Six of the staff have all three types of computing set up, whilst 87 have only one type. Fourteen of the staff have both a laptop and a home computer but no desktop computer on their office desk. All staff who have a laptop computer have a desktop computer either at home or at work.

Construct a Venn diagram to display clearly how many staff are provided with what type(s) of computing setup.

22.By examining the probabilities given in the Venn diagram below, determine whether or not events A and B are independent.

23.If B is an event with P(B) > 0, show that for any event A

P( \B) = 1 P(A\B).

24.Kym, Chris and Lee want to travel together into the bus station in town. Three bus routes pass along their road and, whatever the time of day, they will catch the first bus that comes along. At any given time the probability that bus A comes first is 0.6, the probability that bus B comes first is 0.2 and the probability that bus C comes first is 0.2. All three buses terminate at the same bus station halfway into town.

If Kym, Chris and Lee catch bus A they will always catch bus D for the remaining journey to town. If they catch bus B, the probability that bus D will be first is 0.4 and the probability that bus E will be first is 0.6. If they catch bus C, the probability that bus D comes first is 0.3, the probability that bus E comes first is 0.2 and the probability that bus F comes first is 0.5.

(a)Display the above information in a tree diagram. Include the probabilities for routes AD, BD etc.

(b)Of the possible routes that the three can travel, which is the least probable?

(c)Given that the three catch bus D, what is the probability that they caught bus C for the first stage of the journey?

(d)What is the probability that on their journey into town the three will catch bus B or bus D but not both?

25.A group of students were surveyed as to the type of sports that they liked watching out of football, cricket and rugby. The following responses were obtained:

65% liked watching football;

45% liked watching cricket;

38% liked watching rugby;

21% liked watching all three sports;

16% liked watching exactly two sports;

the number of students who liked watching cricket and football but not rugby was twice the number of students who liked watching cricket and rugby but not football;

5% liked watching only rugby.

(a)Draw a Venn diagram to represent the above information.

(b)If a student is chosen at random from those surveyed, what is the probability that this student likes watching

(i)cricket and rugby,

(ii) cricket and rugby but not football,

(iii) football, given that s/he likes rugby,

(iv) none of these sports?

(c)If there were 20 students who liked watching football and rugby but not cricket, how many students were surveyed in total?

26.Consider two events A and B such that P(A/B) = 0.8, P(B/A) = 0.5 and P(A(B) = 0.81.

(a)Calculate P(A(B).

(b)Show that A and B are not independent.

27.A bookshelf has 3 different dictionaries, 5 different volumes of an encyclopedia and 3 different novels. How many different ways can these be arranged if

(i)there are no restrictions,

(ii) the dictionaries have to be together,

(iii) the dictionaries have to be together and the volumes of the encyclopedia have to be together,

(iv) the dictionaries have to be together or the volumes of the encyclopedia have to be together?

28.Driving along a particular road, Lee has to pass through four sets of traffic lights.

The probability that Lee has to stop at the first set of lights is 0.4.

If Lee had to stop at the first set of lights, the probability of having to stop at the second set of lights is 0.6, otherwise the probability of having to stop is 0.5

If Lee had to stop at both the first and second set of lights, then the probability of having to stop at the third set of lights is 0.65. If Lee only had to stop at the first set of lights, the probability of having to stop at the third set of lights is 0.55, otherwise the probability of having to stop is 0.45.

If Lee had to stop at two out of the first three sets of lights, then the probability of having to stop at the fourth set of lights is 0.7, otherwise the probability of having to stop is 0.4.

Determine the probability that Lee has to stop at two out of the four sets of lights.

29.A group of 100 musicians are attending a party. Between them they can play four instruments.

None of the twenty drummers can play any other instrument.

Ten musicians can each play three instruments.

Half of the non-drummers can play the violin.

Sixty musicians can play the guitar and twenty can play the flute.

Ten of the flautists can play the violin.

Seventy of the musicians can only play one instrument.

Twenty five guitarists can play the violin.

(a)Display this information in a Venn diagram, indicating clearly on your diagram how many musicians fall into each category.

(b)Two people, at random, leave the party. Calculate the probability that

(i)neither is drummer,

(ii)at least one of them can play the violin.

30.There are 100 students in a language class, all of whom speak at least one of the languages, English, French and German. In addition, the following facts are known:

18 speak French only

19 speak at least English and French

52 speak German

15 speak at least English and German

55 speak only one language

21 speak at least French and German

(a)Draw a Venn diagram to represent the above information.

(b)Suppose 4 of the students who only spoke French and 3 of the students who only spoke English learnt to speak German. Now, how many students

(i)speak all of the three languages,

(ii) speak French and German, but not English,

(iii) speak German only,

(iv) speak exactly two languages?

31.A pathology service performs blood tests to detect the presence of a certain type of virus called Virus X. For 5% of blood samples with Virus X, the test suggests its absence, while for 10% of samples without the virus the test suggests its presence. From past data, it is known that 30% of all samples received have the virus. Suppose that one of the fresh samples is taken at random and tested for Virus X.

Calculate the probability that the sample

(a)tests positive,

(b)contains Virus X, given that the test is positive,

(c)tests positive or has the virus in it.

32.A retail outlet commissioned a research team to survey local residents to find the number of homes that use brand As washing machine, fridge and oven. Of the 200 homes surveyed, 24 used all three appliances, 54 used the fridge and oven, 42 used the washing machine and oven, and 24 used none of the three. Furthermore, it was found that exactly half of the surveyed homes used brand As fridge, and exactly half used the washing machine. The probability of a surveyed home using both the fridge and washing machine was 1/5.

Of the surveyed homes, find:

(i)the number that used brand As oven only.

(ii) the number that used exactly one of brand As three appliances.

(iii) the number that used exactly two of brand As three appliances.

(iv) the probability of a home using brand As fridge and oven but not the washing machine.

Three of the surveyed homes are selected at random.

(v) What is the probability that all three homes use brand As fridge?

33.The car registration number plates in a country are required to have exactly six characters, three letters of the alphabet followed by three digits such as

How many different number plates are possible if there are:

(i)no further restrictions?

(ii) no repetitions of either letters or digits?

(iii) three consecutive letters in alphabetical order followed by three consecutive digits in descending order, as in the sample registration plate shown?

34.The following data from the Australian Bureau of Statistics gives the number of nights, in thousands, spent by international visitors to Australia in 1999 by State (first six entries in table ) or Territory.

NSW Vic Qld SA WA Tas NT ACT

38334 20494 24928 4542 12349 1694 3044 2161

For a randomly chosen international visitor night in 1999, calculate the probability that it was

i.spent in WA

ii.spent in one of the Territories (NT and ACT)

iii.not spent in NSW nor Qld.

35.Kim has three favourite CDs : P, Q and R. He always plays two of them whilst completing his homework each night. There is a 50% chance that he will put CD P on first each evening and a 20% chance that he will put CD R on first. At anytime there is a 60% chance that he will follow CD P by CD Q, a 40% chance that he will follow CD Q by CD P and a 70% chance that he will follow CD R by CD P. Kim never plays the same CD twice.

a.Display the above information in a tree diagram, indicating clearly the probabilities for playing each CD and the probabilities for each possible order for playing the CDs

b.What is the probability that Kim will not play CD Q ?

36.a.Complete the table below to show the number of student councillors at

City Community College.

Male

Female

Total

Year 12 6

10

Year 11

5

8

Total

b.Calculate the probability that a randomly selected student councillor

i.is a female

ii.is in Year 12 given that the councillor is male.

c.Three student councillors are selected at random. Find the probability that this selection includes at least one male from Year 12 and at least one female from Year 12.

37.Two events, A and B, in a sample space are such that

P(A) = 0.7, P(B) = 0.4 and = 0.1 [or P(A ( B)' = 0.1].

(a) Find P(A ( B) .

(b) Are the two events A and B independent? Justify your answer.

38.A health centre is open Wednesday and Friday evenings and during the day on Saturday and Sunday. The Venn diagram below represents the number of patrons using the centre in one particular week.

(a) How many patrons used the health centre in the particular week?

(b) During this particular week, what was the probability that a randomly chosen patron used the health centre

(i)every time the centre was open,

(ii)only once,

(iii) on Wednesday and Friday only,

(iv)on Friday and one other time?

39.In a certain year, Mrs Chan taught Calculus in one school and Applicable Mathematics in another school. At the end of the year she summarised the percentage marks (rounded to nearest whole number) obtained by the students in the two classes. This is given in the frequency table below.

% mark range CalculusApplicable

0-1912

20-3932

40-5935

60-79911

80-9957

If a student from one of these classes is picked at random, what is the probability that the student

(a)received a mark greater than or equal to 60,

(b)was in the Applicable Mathematics class,

(c)was in the Calculus class and received a mark ( 80,

(d)was in the Calculus class, if the student had received a mark between 60 and 79 inclusive?

40.Health authorities are dealing with a certain disease in a large community. A cost-effective but not entirely reliable screening test is devised to check the population for disease. It is estimated that 8% of the population have the disease.

The tree diagram below shows some of the associated probabilities (in boxes).

(a)

Complete the tree diagram by writing the appropriate probabilities in the seven empty

boxes. Note: boxes are for values/answers only and are not part of the diagram.

(b)If a person tests positive, what is the probability that he/ she has the disease?

A more expensive (and accurate) test is used to retest all those who tested positive on the first test. This test is accurate 95% of the time (regardless of whether the person does or does not have the disease).

(c)What is the probability that a person who tests positive on both tests has the disease?

41.The following tree diagram represents the probability of Chris

selecting two of his favorite six shirts to wear for the weekend. Some of

the associated probabilities are given.

shirt D 0.4

0.08

0.2

shirt E 0.3

0.06

shirt A

shirt F

0.3

0.06

shirt B

shirt D

shirt C

shirt E

0.42

shirt F

0.28

(a) Fill in the missing values (in the boxes provided) on the tree diagram.

(b) For a randomly chosen weekend, calculate the following probabilities:

i.Chris chooses shirt F,

ii.Chris chooses shirt A given that he also chooses shirt F,

iii. Chris chooses shirt D given that he does not choose shirt B.

42.A group of male and female adults is surveyed about their favorite game.

The results of this survey are summarized in the Venn diagram below. Unfortunately, some of the data is unreadable due to a smudge on the data collection sheet.

S

B

24

x

20

24

4

12

M

M:male

B:likes badminton

S:likes squash

a.Suppose that there are 100 adults in the group surveyed. Calculate the probability that a randomly chosen adult from this group

i.likes badminton and is female

ii.is female or likes badminton

ii.does not like squash given that the adult is female

b.A second group of adults is also surveyed. The data collection sheet also has a smudge on it and the Venn diagram above also summarises the data. Suppose that for this second group the events being female and likes badminton are independent events. Showing all your working, calculate how many adults are in this second group.

R

B

C

A

B

C

A

B

C

A

S

T

R

T

S

R

T

S

AB

0.30.2

0.3

UVW 654

0.068

0.138

_1048064102.unknown

_1048339923.unknown

_1120385083.unknown

_1172411746.unknown

_1048064130.unknown

_1048062077.unknown