The probability that Amad ,i leo w*gk,fgz91y,nda successfully passes her maths exam depends on the exam topic. The probability that the topic is kye,dgol1 gi,z9 ,*fw statistics is 0.35 and the probability she passes this topic is only 0.3. The probability that Amanda passes any other topic during the exam is 0.8.
1.Complete the following tree diagram.




2.Find the probability that Amanda does not pass the exam.   

  Mark wants to meet his friend Jeremy after School to ask him some questions aboh.t ;rb+ ocqd18 el2zyoq -/hgut their Chemistry homework. On a typical school afternoon, Jeremy either goes to the park for some exercise or goes to the library to complete his homework.;o boe1q-ty d c.qh2 8+hglr/z
Jeremy has previously told Mark that he goes to the park two out of every five afternoons each school week.
If Jeremy goes to the park, the probability that Mark will meet Jeremy in the park is 0.2 . If Jeremy goes to the library, the probability that Mark will meet Jeremy in the library is 0.7 .
These scenarios are represented on the tree diagram below.

1. Complete the tree diagram.
2. Find the probability that Mark will meet Jeremy that afternoon.   

  A bag contains 6 white and 4 orange table tennita/ahl4c5fx h:0b/r1ekjrp5s balls. Jack selects a ball at random from the bag and then, after1lxh 5afb5 pk0 ha:ec/ r4/jtrwards, John selects a ball at random from the bag.
1. Complete the tree diagram.



2.Find the probability that John chooses a white ball.   

  During an exam that has a tot1po9 zxoze 0ym:-vm pk u/axk c+,-p2aal of eight questions, Paula correctly solved Questions 3,4,6,7 and 8 . During the sa+9ao2xkoye-pum x/0 , vkm-:zap pc1zme exam, Quentin correctly solved Questions 1,2 3,6 and 8 .
1. Represent this information on the Venn diagram below. P represents the
set of questions correctly solved by Paula. Likewise, Q represents the set of questions correctly solved by Quentin.

2. Find $n(P \cup Q)$ .   
3. Find the probability that a question at random has been solved correctly by Paula or Quentin, but not both.   

  A lego storage box is left with 4 bl ,sg,j0k7ce1ggjyh ,zue and 7 red bricks. Max takes a brick from the box at random and attaches it to his new hjz7j ec1 0gkg,,yg, scastle. He then chooses another brick fron the same box at random.


1.Complete the following tree diagram.
2.Calculate the probability that both bricks chosen by Max are of the same color.≈   

  The probability that a soccer team scores i( shf*9nx3qd jls l6yu7hr +5wn the first half is 0.6 . If the team scores in the first half, the probability that it will score in the second half is 0.3 . If the team does not score in x+l79rd s ljysnu(h*hwqf5 63the first half, the probability it will not score in the second half is 0.2 .
1. Using the information given, complete the following tree diagram.

2.Calculate the probability that the team will score in the second half.   

The following transitioj s,y-rpuq;eh4cb+lhk 5)8 sln diagram reflects the proportions of customers that Qatar Airways loses to its competitor airlines each yea;+ sl)8skq- u,5c bjheyrh pl4r, and vice versa



1. Construct a transition matrix $\boldsymbol{T}$ with elements in decimal form.
2. Interpret the meaning of the elements with values
(1) 0.15
(2) 0.75

Assume that the initial state of the market share is

$s_{0}=\left[\begin{array}{l}
1 \\
0
\end{array}\right]$

3. Determine the market share of Qatar Airways after 5 years.

  There are 3 pairs ofe8tr.c58ndj tpl:(p i white coloured socks and 4 pairs of red coloured socks in a draw. Anne picksct jp .idpt: er8n(58l one sock from the draw at random and puts it on. She then picks another sock from the draw at random and puts it on.
1. Complete the following tree diagram.

2.Find the probability that Anne goes for a walk in the socks of the same color.≈   

  Two bags of marbles contain green a(aj q t1n;w1)u4bkb dhxb el.4ygf* x9cgj15znd orange marbles. Bag A contains 7 green marblk bcbb*45dt);(f e1xwhy 9 j41xuqga. j1lznges and 5 orange marbles, while Bag B contains 3 green marbles and 9 orange marbles.
One marble is selected from Bag A and then Bag B.
1. Determine the probability of selecting two marbles of the same colour.

Both marbles from part (a) are put back into the bags they came from. This time, an unbiased coin is flipped to determine which bag to select a single marble from. If the coin flips heads, a marble is selected from Bag A. If the coin flips tails, a marble is selected from Bag B.≈   
2. Determine the probability of flipping tails and then selecting a green marble.   

A rat is placed in the maze shown in the diagram bee2ix +57hb9wdrfkkiji f5o:6low. The maze has four rooms (A-D) with open doors between the rooms. For example, there is one open door connecting rooms C and D, whereas there are two opebjok+ f7f ew96idk:i25rxh i 5n doors connecting rooms B and D.



Every minute, a buzzer is sounded which prompts the rat to move to a connecting room. The probability of the room the rat chooses to move to is determined by the number of open doors available to it. Each time the rat moves and chooses a door to pass through, the probability of each available open door is equal.
1. Determine the transition matrix describing the room the rat chooses to move to each time the buzzer sounds.

The rat is initially placed in room C.
2. Find the probability of the rat being in room D after 15 minutes.

  Samantha collects a basket of apples, however only 80 \% of+ ; ap0m-y(dpvwk j6le them are ripe. The probabil(ejy0d -mw+pa;vl6p k ity that a ripe apple has a worm inside the apple is p . The probability that an unripe apple has a worm inside the apple is 0.1 .
1. Complete the following tree diagram.


2.Find the value of p, assuming that exactly half of all apples have worms inside.   

  A bag contains 8 blurulue 803 rxytx53v9 re marbles and 4 red marbles.
Ray selects two marbles from the bag in succession, without replacement.
1.Complete the missing probabilities on the tree diagram below.


2. Determine the probability of:
(1) Ray selecting two red marbles.≈   
(2) Ray selecting two marbles of different colour.≈   

  Probability100 students9 6yeul4 chfz6c.)zlb are asked what pets they like from a choice of cats, dogs and hamsters. The res9 lhyecz 64lzb.fuc)6ults are as follows.
60 like cats
64 like dogs
61 like hamsters
34 like cats and dogs
46 like dogs and hamsters
43 like cats and hamsters
32 like cats, dogs and hamsters
1. Represent this information on a Venn diagram.
2. Find the number of students who
(1) like dogs only;   
(2) like both dogs and cats but not hamsters;   
(3) do not like pets.   

A student is randomly chosen from this group of 100 students.
3. Calculate the probability that this student likes only two kinds of pets.   
4. Calculate the probability that this student likes only dogs, given that this student only likes one kind of pet.≈   

Consider the following Venn diagramsfwlf dqo(v289e0oazn 0m xue09l: i z4h*tkv;.



1. Write down an expression, in set notation, for the shaded region represented by
(1) Diagram 1;
(2) Diagram 2;
(3) Diagram 3.
2. Shade on the Venn diagrams the regions represented by the sets
(1) $A^{\prime} \cup B$ ;

1. Write down an expression, in set nhqa*f xp s26znp8cz1r6n/rf(8 xbs8zotation, for the shaded region represented by raf2/ x6q bn8pxp*zhnrs(z 88s16fcz
1. Diagram 1;
2. Diagram 2;
3. Diagram 3.
2. Shade on the Venn diagrams the regions represented by the sets
1. $A^{\prime} \cup B$ ;Three Internet Service Providers (ISPs) are available in a small town. During the year, ISP A is expected to retain 85 \% of its customers; 10 \% will be lost to ISP B and 5 \% to ISP C. ISP B is expected to retain 80 % of its customers; 10 % will be lost to each of the other two ISPs. ISP C is expected to retain 75 % of its customers; 15 % will be lost to ISP A and 10 % to ISP B.
1. Write down a transition matrix that describes the exchange of market shares between the three ISPs during the year.

The current market share held by ISP A is 0.2 , by ISP B is 0.3 and by ISP C is 0.5 .
2. Find the market share held by each ISP after one year.
3. Find the market share held by each ISP after five years if the same trend of market share exchanges between the three ISPs continues.

  At the local supermarket, out of the 100 cartons of eggs ony:f, dbhq4os/+lp -qo sale, 8 have at least one egg insf-+hpbd/qos4 q,o l y:ide that has cracked.
If two cartons are chosen at random from the 100 without replacement, find the probabilities of:
1. Neither cartons having a cracked egg inside.≈   
2. Both cartons having at least one cracked egg inside.≈   
3. One carton not having a cracked egg inside, while the other having at least one cracked egg inside.≈   

  125 sports fans were interviewed a9k0v;nl uhvg b t5r/,ond asked what types of sport they have been to in the last year from a cno5/rbv9 t vu0lh,;kg hoice of soccer (S) , rugby (R) or baseball (B) . The following information was obtained.
74 had been to soccer
36 had been to rugby
66 had been to baseball
20 had been to soccer and rugby
27 had been to soccer and baseball
24 had been to rugby and baseball
15 had been to all three types of sport
1. Draw a Venn diagram to show the above information.
2. Find the number of sports fans who, in the last year, had been to
1. baseball only;   
2. both rugby and baseball but not by soccer;   
3. at least two types of sport;   
4. none of the three types of sport.   

A sports fan is selected at random from those who were interviewed.
3. Find the probability that the sports fan had been to only one type of sport in the last year.   
4. Given that the sports fan had been to only one type of sport in the last year, find the probability that the sports fan had been to rugby. ≈   

  50 people waiting in line at a movie cinema were asked their preference of movie n dhz/*:i,nb ubnch:1 genres: action (A) , comedyd cb1n/:zun:b*h i,h n (C) or drama (D) .
5 prefer all three
12 prefer action and comedy
7 prefer action and drama
9 prefer comedy and drama
8 prefer action only
3 prefer drama only
1. Draw a Venn diagram to represent the information provided so far.
2. Write down the number of people who prefer action but not drama.

There are 22 people in total who prefer comedy.   
3. 1. Calculate the number of people who prefer comedy only.   
2. Find the number of people who prefer none of these three genres.

A person is chosen at random from the 50 people that were waiting in line.   
4. Find the probability that this person
1. prefers action;   
2. prefers comedy and drama but not action;   
3. does not prefer either action or comedy;   
4. does not prefer action given that the person does not prefer comedy. ≈   

The Venn diagram below shud,z xt wto,-x tlh gh(49//wbows the sets A,B,C and U


Determine whether the given regions are completely shaded.
1. $A^{\prime}$ ;
2. $A \cap B$ ;
3. $B \cup C$ ;
4. $A^{\prime} \cap C$ ;
5. $(A \cup C)^{\prime}$ ;
6. $A \cap B \cap C$ .

Two competing radio stations, station A and statiom-l0wx 8ou2 ton B, each have 50 \% of the listener market at some point in time. Over each one-year period, station A manages to take awa8t l20 o-wouxmy 15 \% of station B's share, and station B manages to take away 10 \% of station A's share.
1. Write down a transition matrix that describes the exchange of market shares between the two stations over each one-year period.
2. Find the market share held by each station after one year.
3 . Write down the market shares of stations A and B over a five-year period.
4. Find the market share held by each station in the long term if the same trend of market share exchanges between the two stations continues indefinitely.

  A group of international stude*h ;kksq/ph -5lpedf*4 fqee(nts at a college were surveyed. They were asked which foreign languages they spoke at a conversational l(pqhhfk*d * 4;efsqe-l kep/5evel.
The following results were received:
22 spoke Spanish
35 spoke French
5 spoke Chinese
6 spoke Spanish and French
2 spoke French and Chinese
2 spoke Spanish and Chinese
1 spoke Spanish, French and Chinese
2 students spoke Chinese only
7 students did not speak any foreign languages
1. Represent this information on a Venn diagram. Denote with S, F , and
C the sets of students who spoke Spanish, French, and Chinese respectively.
2. Find the number of students who were surveyed.   
3. Shade the set S $\cap F \cap C^{\prime}$ .
4. Find $n\left(S \cap F \cap C^{\prime}\right)$ .   
5. A student is chosen at random from those surveyed. Calculate the probability that this student spoke
1. Spanish; ≈   
2. Chinese, given that this student did not speak French;   
3 . at least two foreign languages.≈   
6. Two students who were surveyed are chosen at random. Calculate the probability that both students spoke Spanish.≈   

  Adam has an unbiased 06kld4rak3+ chl;ul7g-kywew x b)id u0d9 x5 octahedral (eight faced) die which has the numbers 1,2,3,4,5,6,7,8 . h7xl 6dwkdi ul-yrku )0cw kb0xde9 a 5l;+3g4
Bella has an unbiased pentahedral (five faced) die which has the numbers 1,3,5,7,9 .
1. Complete the Venn diagram with the numbers written on Adam's die (A) and Bella's die (B) .

2. Find $n\left(A^{\prime} \cup B^{\prime}\right)$ .

Adam and Bella are each going to roll their die once only. Charlie says the probability that each die will show the same number is $\frac{1}{10}$ .   
3. Determine whether Charlie is correct. Give a reason.

  A group of chemistry students at Ind 9f 5g4pbmncw3iana University Bloomington were surveyed. They were asked which of the following textbook formats: e-textbook ( cfmg9 p 45bnw3 E ), paperback (P) or hardcover (H) , they used in their courses.
The following results were obtained:
108 students used paperback
64 students used hardcover
45 students used paperback and hardcover
68 students used e-textbook and paperback
49 students used e-textbook and hardcover
35 students used all three textbook formats
18 students used e-textbook only
15 students used none of the three textbook formats
1. Use the above information to complete a Venn diagram.
2. Calculate the number of students who were surveyed.   
3. 1. On your Venn diagram, shade the set $(E \cup P) \cap H^{\prime}$ .
2. Find $n\left((E \cup P) \cap H^{\prime}\right)$ .   
4. A chemistry student who was surveyed is chosen at random. Find the probability that
1. the student used an e-textbook; ≈   
2. the student used a hardcover, given that they had not used an e-textbook;   
3 . the student used at least two of the textbook formats.

The Chemistry Department at IU Bloomington has 2560 students.   
5. Find the expected number of students in the Chemistry Department that used e-textbooks. ≈   

  At a travel agency, a survey conducted on the 32 age/ rwasmzm,zi ;oar9wi rj(/0. nts shows that 18 agents can speak Spanish fluently, 12 agents speak French fluently 0za (/oz;9/iwai.mr r, jwsrm and 8 neither Spanish nor French.
1. Find the number of agents who can speak both Spanish and French fluently.
2. Find the probability that an agent chosen at random from this travel agency speaks exactly one of these two languages fluently. ≈   

  A seafood restaurant conducts a promotional campaign dure6z9yvv lzt* -ing the month of April. Over the 30 days, there will be 15 days wily* -9ztvv6ze th a 50 \% discount on drinks and 10 days with a 50 \% discount on meals. However, on 8 of the days there will be no discount on either drinks or meals.
1. Calculate the number of days there will be a 50 % discount for both drinks and meals.
2. Find the probability that, for a guest coming to the restaurant for dinner on a random day in April, the guest will receive a 50 \% discount on either drink or food, but not both.≈   

Austin allocates a portion of i3obf-)t /2 8svaii hcxd6 ktc:7 ksz6his employment salary each month to investing and invests this money into two stock funds: A and B. He adjusts his investment portfolio each month according to the following tradb/o )a sct6:ihvckz 2-8isi3 ktx76fnsition diagram.



1. Construct a transition matrix $\boldsymbol{T}$ with elements in decimal form.
2. Interpret the meaning of the elements with values
1. 0.1
2. 0.7

The initial state of his investment portfolio is 100 % in stock fund B.
3. 1. Find the investment proportion in stock fund A after 3 months.
2. Determine the long term steady state proportion of his investment between the two stock funds.

  In a college lottery, there are tf),hx y7o fwy/t1l iol-3o, evwo types of tickets: 'standard' tickets and 'luck1 yw/hovyxfl,t 3i)oo,7-e lfy' tickets. Every student is given one ticket at random. The total number of the standard tickets is four times greater than the number of lucky tickets.
The probability of winning the lottery with a standard ticket is $ \frac{1}{100} and \frac{1}{40}$ for a lucky ticket. The tree diagram below illustrates this.

1. Calculate the probability that a randomly chosen student will win the lottery.   
2. Calculate the probability that a randomly chosen student will get a lucky ticket and still lose the lottery.   
3. Calculate the probability that a randomly chosen student, given they won the lottery, had a standard ticket. ≈   

  Jennifer is about to d5p0b gmr2j hl3:xj5w take three exams in order: Mathematics, Chemistry and Biology. Based on what5pmjj50wb lx d3 :r2hg she has learned and practiced, she estimates that the probabilities that she will pass Mathematics, Chemistry and Biology exams are 0.8,0.6 and 0.7 , respectively. Assuming that the probability of Jenifer being passing an exam is independent of the others, find the probability that
1. she will pass all three exams;   
2. she will pass only one of the exams;   
3. she will pass the last two exams given that she doesn't pass the first exam;   
4. she will pass at least one exam.   

  A group of 80 students arr(qzm,bp0u 1 le asked which books they read from a choice of fiction books, non-fiction books and autobiographiesz,0q lrub1p(m . The results are as follows.
5 read only autobiographies
8 read only fiction books
10 read only non-fiction books
12 read autobiographies and fiction books, but not non-fiction books
27 read autobiographies and non-fiction books, but not fiction books
4 read non-fiction books and fiction books, but not autobiographies
x read all three kinds of books
1. Represent this information on a Venn diagram.
2. Find the value of x , if one student does not read any of these book types.   
3. A student is randomly chosen from this group of 80 students. Calculate the probability that this student reads
1. non-fiction books;   
2. fiction books;   
3. autobiographies, given that this student reads non-fiction books. ≈   
4. Determine whether the events in part (c)(i) and part (c)(ii) are independent. Give a reason. ≈   

  A group of 150 students are asked w e owpo/t3o)7q:/yblwhich beverage they prefer from a choice of coffee, tea and juice. The results aryw ):7 qo3/wt/oobpele shown below.
4 like only tea
2 like only juice
1 like none of these beverages
35 like coffee and juice
x like tea and coffee
30 like all three beverages

The number of students that like coffee and juice only is half the number who like coffee only.

1. Include all information provided so far on the Venn Diagram above.
2. The number of students that like tea and coffee only is equal to the number of students that like tea and juice only. Find the value of x .   
3. Find the number of students who like
1. tea and juice;   
2. tea or juice.   
4. A student is randomly chosen from the 150 students. Calculate the probability that this student likes
1. all three beverages;   
2. only two beverages; ≈   
3. juice, given that she likes coffee. ≈   
5. Two students are randomly chosen from the 150 students. Calculate the probability that these two students like all three beverages. ≈   

  The delivery time (in minutes) for a pizza frknv sm36g , qx.0tp+cnom Pizzeria da Michele follows a normal distribution with a mean of 30 and standard deviation of 5 . On a particular day, 12 \% of the pizzas were delivered in more th3.psn6vtx nc,0 gmqk+ an k minutes.
1. Find the value of k .

As an advertising campaign, Pizzeria da Michele is offering customers a free pizza coupon if their pizza is not delivered in 37.5 minutes or less. ≈    min
2. Find the probability that a randomly chosen customer will get a free pizza coupon. ≈   
3. Given that Vito gets a free pizza coupon from Pizzeria da Michele, find the probability that his pizza was delivered in 40 minutes or more. ≈   

The vegetables sold at supehdl-(n9 cp c,zrmarkets in a town are supplied by three major retail suppliers: A, B and C. According to an analysis report, supplier A retains 80 % of their customers each year and cl (hd,p-n z9close 15 % to supplier B and 5 % to supplier C. Meanwhile, supplier B retains 70 % of their customers each year and lose 20 % to supplier A and 10 % to supplier C. Supplier C retains 75 % of their customers each year and lose 10 % to supplier A and 15 % to supplier B.
The report also shows that suppliers A, B and C currently hold a market share of 50 %, 25 % and 25 % , respectively.
1. Find the market share held by each supplier after three years.
2. Determine the steady state market share held by each supplier if the same trend remains unchanged.

Laura creates a list of her favorite songs that includes threpv9/mw .mu (9qux-f *o z-rwane genres: Jazz, Slow Rock and Country. After her current song ends she randomly selects the next song and the probabilities of genre of the next song arpzu m(fn v ./r*aum9-9xwqo -we outlined in the following table.



Laura starts her day with a Slow Rock song and is now listening to her fourth song.
1. Determine the genre of music she is currently most likely listening to.
2. Determine which genre of music she listens to most over the long term.

  A mathematics teacher designed a new type of probabilifo5 5zjvnxc)vk) z3rm t::y01hxyk -ity game to play with her students. In the game, a student d1zj:kn m-x0y3v5x)i 5fo: rytc )zkh vraws two marbles at random from a bag in succession, with replacement. The bag contains 8 marbles: 1 red, 3 white, 2blue and 2 green. The points scored for each coloured marble is shown in the table below.



The score the student receives in the game is the sum of the points from the two draws. This is illustrated by the sample space diagram shown below.


1. Determine the value of c in the sample space diagram above.

Lin plays the game once. Let the random variable L represent Lin's score. =   
2. Using your answer in part (a), complete the missing column in the following probability distribution table for L .$\frac{a}{b}$ a =    b =   




3. Find the probability that
1. Lin scores at least 4.$\frac{a}{b}$ a =    b =   
2. Lin scores exactly 6 , given that she scores at least 4 .$\frac{a}{b}$ a =    b =   
4. Find Lin's expected score in the game.$\frac{a}{b}$ a =    b =   

  The table below shows the distribution of the number of fish caught during a fisgv;vz-sl eyt :z +;f*lhing contest by 100 contestants.t-:+gyfv;z z;llse v*




1. Calculate
1. the mean number of fish caught by a contestant; ≈   
2. the standard deviation. ≈   
2. Find the median number of fish caught in the contest.   
3. Find the interquartile range.   
4. Determine if a contestant who caught 10 fish would be considered an outlier.

One of the contestants is chosen at random.(a,b) a=   b=  
5. Find the probability that this chosen contestant caught 6 fish or more.

A second contestant is randomly chosen.   
6. Given that the first contestant caught 6 fish or more, find the probability that each of these two contestants caught exactly 7 fish.

The cost of the fishing equipment used by the 100 contestants is normally distributed with a mean of 1200 US Dollars (USD) and a standard deviation of 300 USD. ≈   
7. 1. Calculate the probability that a contestant chosen at random spent at least 1000 USD for fishing equipment. ≈   
2. Calculate the expected number of contestants that spent at least 1000 USD on fishing equipment.   

  The first two characters of a student ID nu 1dpo uiou,vz+. 0blcw wf8,(pmber is composed by the following rules:pof(u180vozw +u icl, w,.pb d
- one letter is chosen from student's first name at random;
- one digit is chosen from student's date of birth at random (dd/mm/yyyy).

Bob Stewart was born on the 12th of January, 2001(12/01/2001). Find the probability that
1. Bob's ID starts with "B1".   
2. Bob's ID starts with "B" or its second character is "2".   
3. Bob's ID starts with "O" given that the second character is not 0 . ≈   

  A jar of candy contains 14 sweet pkimu stw(l6 b;41aw5i o7le3aieces and 8 sour pieces. Sarah selects one piece at random and eats it. This piece Sarah selected was sweet. ai7l;6ws45o uem(aw3 b1kltiThe tree diagram below represents the outcomes for Sarah, given this first selection.



1. Determine the values of:
1. a $\frac{a}{b}$ a =    b =   
2. b $\frac{a}{b}$ a =    b =   
3. c $\frac{a}{b}$ a =    b =   
2. Determine the probabilities of:
1. Sarah selecting two sweet pieces of candy in a row. ≈   
2. Sarah selecting two different types of candy.

A second jar of candy contains only sweet pieces, 15 of which are yellow and 11 are blue. Charlotte selects two pieces of candy from this new jar at random, without replacement. Determine the probabilities of: ≈   
3. 1. Both pieces being blue. ≈   
2. Both pieces being the same colour. ≈   
3. The second piece being yellow, given the first piece was blue.   
4. If Charlotte didn't like the yellow flavour and kept selecting (and removing) pieces at random until she selected the one blue piece she wanted, calculate the probability of 5 pieces being selected in total. ≈   

  The Future House factory produces06h u85v6 zi cqiyd8zo54jcgg LED light strips, 5 \% of which are found to be defective. q054zyioigud65 g68 hvcc8zj
1. Write down the probability that a light strip produced by Future House is not defective.

Jack buys two light strips produced by Future House.   
2. 1. Find the probability that both light strips are not defective. ≈   
2. Find the probability that at least one of Jack's light strips is defective.

The Kingstar Lighting factory also produces LED light strips. The probability that a light strip produced by Kingstar Lighting is not defective is p . John buys three light strips produced by Kingstar Lighting.   
3. Write down an expression, in terms of p , for the probability that at least one of John's three light strips is defective.

  106 runners at a marathh.(7 dbd x5h i*odydl7zm-j/m on event were asked through which media channels they received information about the marathon. The summary shows that 52 runners answered "TV advertising", 64 answered "social media" and 14 answered "others". Find the probability that a runner selected at random from the marathon received informat bd(75-zi/ohdl *.y h7xjdmd mion about the marathon through:
1. both TV advertising and social media; ≈   
2. only TV advertising. ≈   

  The table below shows the distribution of trip numbers made by a group ofjhja kep)i2d.w: /r1k 100 taxi drivers sue. )ai1jh:dkw/ pj2krrveyed on a working day in Sydney.


1. Find
1. the mean number of trips made by the taxi drivers;≈   
2. the standard deviation of the number of trips made.≈   
2. Find the median number of trips made by taxi drivers.   
3 . Find the interquartile range.
A taxi driver is chosen at random from the group of 100 taxi drivers.   
4. Find the probability that this taxi driver made 13 or more trips.

A second taxi driver is chosen at random from the group of 100 taxi drivers.   
5. Given that the first taxi driver chosen at random made 13 or more trips, find the probability that both taxi drivers made 14 trips.

The amount of time that the 100 taxi drivers waited for their next client was normally distributed with a mean of 15 minutes and a standard deviatior of 4 minutes.≈   
6. 1. Calculate the probability that a taxi driver chosen at random waited at least 12 minutes for the next client.≈   
2. Calculate the expected number of taxi drivers that waited at least 12 minutes for their next client.

The 100 taxi drivers were selected for the survey by ordering taxi identification numbers in ascending order, then selecting every 10 th number.   
7. Identify the sampling technique used in this sampling method.

  The table below shows the breed of cats exhibited atoni mo kr 9.r)qhek.7: the first widely known cat show in London at )9hqo7ki .mk nreo: .rCrystal Palace in 1871. In this early exhibition, only sixty-five cats were on displace.



1. For a cat in this exhibition chosen at random, find the probability that
1. it is a male; ≈   
2. it is a Persian female; ≈   
3. it is Siamese given that it is a male.   
2. Find the probability that two randomly chosen British Shorthair cats are both females.

A $\chi^{2}$ test is carried out at the 5 $\%$ significance level for the data in the table. ≈   
3. State the null hypothesis for this test.
4. Find the expected frequency of female Persian cats. ≈   
5 . Write down the number of degrees of freedom for this test.
6. Using your graphic display calculator, find the $\chi^{2}$ statistic for this test.≈   

The critical value for this test is 15.5 .$x^2$ =   
7. State the conclusion for this test. Give a reason for your answer.

  Applicants to a special division in the Australian Military Forces 7krw;pt (h s;dneed to undertake a fitness test with five components: Sprint, Endurance Run, Push-Ups, Pull-Ups and Squats. Their fitness level on each component is classified as either "At Standard" or "Below Standard". To pass the test, 7 spdt(k ;h;wrapplicants must perform at an At Standard fitness level in each of the five components. In increase intake this year, it was decided that applicants were allowed to omit any one of the five components to increase their chances to pass the test.

The table below shows the number of At Standard and Below Standard performances in each component.

1. A military officer assesses the results and chooses a performance at random. Find the probability that this randomly chosen performance is
1. a performance in Sprint;   
2. a Below Standard performance in Sprint;   
3. a Below Standard performance, given that it is a performance in Sprint. ≈   
2. The military officer groups the performances by component and chooses three performances in Squats. Find the probability that all three are At Standard fitness level.

A $\chi^{2}$ test is carried out at the 5 $\%$ significance level for the data in the table. ≈   
3. State the null hypothesis for this test.
4. Show that the expected frequency of Below Standard performances in Sprint is 8 .
5. Write down the number of degrees of freedom for this test.   
6. Use your graphic display calculator to find the $\chi^{2}$ statistic for this data.

The critical value for this test is 9.488 .$x^2$ ≈   
7. State the conclusion of this $\chi^{2}$ test. Give a reason for your answer.

  A university library has two sections, one for books and thexarinjg:hd 2 dsj9(d*+31 sp- ypwbn* other for journals. The probability that a randomly observed visitor accesses the books section is 0.85 , and the probability that the visitor accesses the journals sect:sn jd1rg*d9+ xdn-2 bp( is3jayh* wpion is 0.3 . Assume that in each visit, visitors access at least one of the two sections.
1. Find the probability that a randomly chosen visitor
1. Accesses both the books and journals section,   
2. Only accesses the journals section.

On a typical day, 120 visitors visit the library.   
2. On a typical day, find
1. The expected number of visitors to access the books section,   
2. The probability that more than 40 visitors access the journals section.

It is found that 70 % of the visitors to the library are students, and that 82 % of these students access the book's section when they visit the librar.≈   
3. 1. A library visitor is chosen at random. Find the probability that the visitor is a student and accesses the book's section.   
2. A non-student visitor is chosen at random. Find the probability that the visitor accesses the book's section.   

After school, a group of six students play a soccer passing game. Alex, Belj)z6* ih0gfjm la, Cleo, Dixie, and Emmy stand in a circle and pass the ball to each other while)i zh6jfg *jm0 Ben, standing in the middle, tries to intercept the passes.

The following diagram shows the possible paths that the ball can be passed between the players, in the form of a directed graph. Some of the students are more likely to pass the ball to their friends than to other students. The paths shown by dotted lines represent a pass that is twice as likely as a pass shown by a solid line. For example, Dixie can pass the ball to Alex and Cleo with probability 0.25 and to Emmy with probability 0.5. Dixie won't pass the ball to Bella.



It is assumed that each player keeps the ball for a constant time before passing it. At the start of the game, Alex has the ball.
1. Determine the transition matrix for the graph.
2. Calculate the probability that Cleo has the ball after exactly four passes have been completed, assuming that Ben has not intercepted a pass.
3. If the players continue passing indefinitely, without an interception, determine which player will spend the least amount of time with the ball.

  A discrete dynamical system /h9 mi;uk)n e k8uduv8is described by the following transition matrix, $\boldsymbol{T}$ ,

$\boldsymbol{T}=\left[\begin{array}{ll}
0.3 & 0.8 \\
0.7 & 0.2
\end{array}\right]$


The state of the system is defined by the proportions of population with a particular characteristic.
1. Use the characteristic polynomial of $\boldsymbol{T}$ to find its eigenvalues.$\lambda_{1}=$    , $\lambda_{2}$=   
2. Find the corresponding eigenvectors of $\boldsymbol{T}$ .$x_1$ = $\left[\begin{array}{l}
a \\
b
\end{array}\right] $ a =    b =    $x_2$ = $\left[\begin{array}{l}
a \\
b
\end{array}\right] $ a =    b =   
3. Hence find the steady state matrix s of the system. s = $\left[\begin{array}{l}
a \\
b
\end{array}\right] $ a =    b =   

  A biologist conducts an experiment to study the pollinationje :nyz //,bx)g7 jvhi preference of bumblebees' on different floral types. In a flight cage, 240 bj:vj i/exg/)y bhn,7z umblebees are free to choose between two species of floral: A. majus striatum or A. majus pseudomajus. The changes of pollination behaviors of these bumblebees after every minute are reflected in the following table.


Initially, 150 bumblebees choose A . majus striatum and 90 bumblebees choose A . majus pseudomajus.
1. Write down the initial state $\boldsymbol{s}_{0}$ and the transition matrix $\boldsymbol{T}$ .$\boldsymbol{T}=\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right]$ a =    b =    c =    d =    $s_{0}=\left[\begin{array}{c}
a \\
b
\end{array}\right]$ a =    b =   
2. Determine $\boldsymbol{T} \boldsymbol{s}_{0}$ and interpret the result. $Ts_0$ = $=\left[\begin{array}{l}
a \\
b
\end{array}\right]$ a =    b =   
3. Find the eigenvalues and corresponding eigenvectors of $\boldsymbol{T}$ . $X_1$ $=\left[\begin{array}{l}
a \\
b
\end{array}\right]$ a =    b =    $X_2$$=\left[\begin{array}{l}
a \\
b
\end{array}\right]$ a =    b =   
4. 1. Write an expression for the number of bumblebees choosing to pollinate on A . majus pseudomajus after n minutes, n $\in \mathrm{N}$ .
2. Hence find the number of bumblebees choose to pollinate on A . majus pseudomajus in the long term.   

  An information technology (IT) company offers paidvwnekdzysy k8hg2/jbb d-re cr49:0j: ;.9e v travelling vacation to its 160 employees every year. The employees can choose between travelling domestically or internationally. It is observed that 50 % of the employees who choose to travel domestically one year, choose internationally the next year. Conversely, 30 % of those who choose to travel internationally one year change to travel domestically tv0jj v.rg4w ek yks/ycb9een9dh2r:zd ;b- :8he following year. For this year, 80 employees chose travelling domestically and 80 employees chose travelling internationally.
1. Write down the initial state $\boldsymbol{s}_{0}$ and the transition matrix $\boldsymbol{T}$ .$\boldsymbol{T}=\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right]$ a =    b =    c =    d =    $\boldsymbol{s}_{0}=\left[\begin{array}{l}
a \\
b
\end{array}\right]$ a =    b =   
2. Determine $\boldsymbol{T} \boldsymbol{s}_{0}$ and interpret the result. $Ts_0$$\boldsymbol{s}_{0}=\left[\begin{array}{l}
a \\
b
\end{array}\right]$ a =    b =   
3. Find the eigenvalues and corresponding eigenvectors of $X_1$=$\boldsymbol{T}$ .$\boldsymbol{s}_{0}=\left[\begin{array}{l}
a \\
b
\end{array}\right]$ a =    b =    $X_2$=$\boldsymbol{T}$ .$\boldsymbol{s}_{0}=\left[\begin{array}{l}
a \\
b
\end{array}\right]$ a =    b =   
4. 1. Write an expression for the number of employees who choose travelling internationally after n years, n $\in \mathrm{N}$ .
2. Hence find the long term steady state number of employees to choose to travel internationally.   

Two grocery stores, store A and store Bn4g 9w1iqgsa l*ur1x., serve in a small city. Each year, store A keeps 30 % of its customers while 70 % of them switch to store Store B keeps 60 % of it9ag rq 1g1w. in*xusl4s customers while 40 % of them switch to store A.
1. Write down a transition matrix $\boldsymbol{T}$ representing the proportions of the customers moving between the two stores.

At the end of 2019 , store A had 8360 customers while store B had 6820 customers.
2. Find the distribution of the customers between the two stores after two years.
3. 1. Show that the eigenvalues of $\boldsymbol{T}$ are $\lambda_{1}=1$ and $\lambda_{2}=-0.1$ .
2. Find a corresponding eigenvector for each eigenvalue from part (c) (i).
3. Hence express $\boldsymbol{T}$ in the form $\boldsymbol{T}=\boldsymbol{P} \boldsymbol{D} \boldsymbol{P}^{-1}$ .
4. Show that

$\boldsymbol{T}^{n}=\frac{1}{11}\left[\begin{array}{ll}
4+7(-0.1)^{n} & 4-4(-0.1)^{n} \\
7-7(-0.1)^{n} & 7+4(-0.1)^{n}
\end{array}\right]$

, where n $\in \mathbb{Z}^{+}$ .
5. Hence find an expression for the number of customers buying groceries from store $\mathrm{A}$ after n years, where n $\in \mathbb{Z}^{+}$
6. Verify your formula by finding the number of customers buying groceries from store A after two years and comparing with the value found in part (b).
7. Write down the long-term number of customers buying groceries from store A.

A city has two major secpvna : zv32428 xrffpk mi1+oq i8onm+urity guard companies, company $\mathrm{A}$ and company $\mathrm{B}$ . Each year, 15 % of customers using company $\mathrm{A}$ move to company $\mathrm{B}$ and 5 % of the customers using company B move to company A. All additional losses and gains of customers by the companies can be ignored.
1. Write down a transition matrix $\boldsymbol{T}$ representing the movements between the two companies in a particular year.
2. 1. Find the eigenvalues and corresponding eigenvectors of \boldsymbol{T} .
2. Hence write down matrices $\boldsymbol{P}$ and $\boldsymbol{D}$ such that $\boldsymbol{T}=\boldsymbol{P} \boldsymbol{D} \boldsymbol{P}^{-1}$ .

Initially company A and company B both have 3600 customers.
3. Find an expression for the number of customers company A has after n years, where n $\in \mathbb{Z}$ .
4. Hence write down the number of customers that company A can expect to have in the long term.

Zoologists have been collecting data about the migration habits of a pl lqgzsi4 u8j ca/. h1(3cd2qfarticular species of mammals in two regions; region X and region Y. Each year 30 % of the mammals move from region X jqgu4lz .1a3qlfs2i(h c8c d/to region Y and 15 % of the mammals move from region Y to region X . Assume that there are no mammal movements to or from any other neighboring regions.
1. Write down a transition matrix $\boldsymbol{T}$ representing the movements between the two regions in a particular year.
2. 1. Find the eigenvalues of $\boldsymbol{T}$ .
2. Find a corresponding eigenvector for each eigenvalue of $\boldsymbol{T}$ .
3. Hence write down matrices $\boldsymbol{P}$ and $\boldsymbol{D}$ such that $\boldsymbol{T}=\boldsymbol{P D} \boldsymbol{P}^{-1}$ .

Initially region $\mathrm{X}$ had 12600 and region $\mathrm{Y}$ had 16200 of these mammals.
3. Find an expression for the number of mammals living in region Y after n years, where n $\in \mathbb{Z}^{+}$ .
4. Hence write down the long-term number of mammals living in region Y .