The probability that Amanda successfully passes her maths exam depends on tbuthdbeg+:*w* i, /gy he exam topic. The probability that the topic is statistics is 0.35 and the probability she passes this topic is only 0.3. The ht gedyibg+** b:w/ ,uprobability 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 Sccw44(pou: pi nhool to ask him some questions about their Chemist pc4o:4 iwu(npry homework. On a typical school afternoon, Jeremy either goes to the park for some exercise or goes to the library to complete his homework.
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 tennis bbyr3 4y-nl9r3uza* 7 anrf,9)csmpoc alls. Jack selects a ball at random from the bag and then, afterwards, John selects a ball at random from the bag. oy)u99n3 * c-,c4bnzs rapm ry7 3lfra
1. Complete the tree diagram.



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

  During an exam that has a total of eight p ,iofik ,b)3-xgwd yaink: 62questions, Paula correctly solved Questions 3,4,6,7 and 8 . During the same exam, Quentin correctly solved Questions 1,2 3,6 a,io a) :,i3ygkfipdb-k x2w6nnd 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 blue and 7 red bricks. Max takes a brick frv g.ph/zo puh556+k icea(rh3om the box at random and attaches it to his new castle. He then chooses aicopa h6zu+hr5g h 3e(v.k5/ pnother 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 soccenzc5e o)e**j/+lt hl d2lb1dzr team scores in the first half is 0.6 . If the team scores in the first half, the probability that it will scez h*lj 1bd5lt*+ nc )ezodl/2ore in the second half is 0.3 . If the team does not score in the 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 transition diagram reflects the-5cokkcv ; bk75 2yyz)gu)a zb proportions of customers that Qatarc 5g)oak2yukz)7 y kvb c;-bz5 Airways loses to its competitor airlines each year, 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 of white coloured socks and 4 pairs*x, b ndgi 1 2r+fll5*rmrtk1e of red coloured socks in a draw. Anne picks one sock from the draw at random and puts it on. She th*r+*2rf, b dirlet1g 1m5 nxlken 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 and orange marx7; rd 5l6a p4r hrr;wd.fd2ka dk(p7rbles. Bag A contains 7 green marbles and 5 orange marbles, while Bag B contains 3 green mar2x7 74kr. ppr6dawr(f k ;5rdha;rlddbles 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 thj93 a.z+d6l7edzfc 4.ko+ouu dsa1 gx e maze shown in the diagram below. The maze has four rooms (A-D) with open doors between the rooms. F36 + cl.ujga.keoxus9 odf14zza+d7 dor example, there is one open door connecting rooms C and D, whereas there are two open 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 apple o*rh;lqn9ck +s, however only 80 \% of them are ripe. The probability that a ripe apple has a worm 9*qnhl+ ocrk; 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 blue marbles and 4 red marbl8c: )nsu 5ec9jajr-+j-j 8eeita er+ mes.
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 students are asked what pets they like 7 1+ge +wkfzwt6yin(bfrom a choice of cats, dogs and hamsters. The results are as follows. g+z(+nwk ibe67tfwy 1
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 diagramssk)7(* ;. gd ej(q ok7jtnhovb;kjg7g.



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 notation, for the xsez kutt 29 ee-b h1p4j6th2-shaded region represented byst 42ehtke-1x96j - h2b euzpt
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 t,-ctkln0ik3fg;6m ew:lz t+ lhe 100 cartons of eggs on sale, 8 have at least one eg n6tll:ie 0tflm3z,+ckg wk;- g inside 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 and asked what type 3psh/u iy.i/ts of sport they have been to in the last year from a choice of soccer (sy/ti i3p.u/hS) , 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 a0nu(c2wr jc tar/n0g4t a movie cinema were asked their preference of movie genres: action (A) , comedy (C) or drama (D r0twnrgc4(/uca 0nj2) .
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 shows the sets bb+ -y.w.l5 vhknc l5kA,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 station B;+lugy:1hf dk:0wi z atd )x:y, each have 50 \% of the listener market at some point in time. Overi:;ly)y h+0k1axd tzfud: wg: each one-year period, station A manages to take away 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 students at a college wer-74g -yp iycuiu5xy *qe surveyed. They were asked which foreign languages they sxgy7u-uyip*i yc45- qpoke at a conversational level.
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 octahedral (eight faced) ;a 3q ei ;b)phfz*5yvb v qig/bkn/i98die which has the numbers 1,2,3,4,5,iay 9/pvnqbfhq3 5bie8v /i*;g)zkb ;6,7,8 .
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 Indiana University Bloomington were surveyed. n, fi-+ hxf7ogThey were ih x,7f on-gf+asked which of the following textbook formats: e-textbook ( 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 32kf4/eced. pyrt / j(x3idqyw5/ afa0 0 agents shows that 18 agents can.rdit0aex4df/ a5/w y cpe j0yk/fq3( speak Spanish fluently, 12 agents speak French fluently 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 campai vmbm,4ata5: ign during the month of April. Over the 30 days, there will be 15 days with 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 t54mm,ibv:a a 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 his employment j/vqr :;u nlej*mvo .,salary each month to investing and invests this money into two stock funds: A and B. He adjusts his investment portfolio each monthjnl uj/oq.*,:;rv me v according to the following transition 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 two types of tickets: 'standard' tickets and5th6m t.niivm5 /yn0m 'lucky' tickets. Every student is given one ticket at random. The total numbvn5/tthiym60. ni m5mer 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 take three exa5 ux;:nyegx0k ms in order: Mathematics, Chemistry and Biology. Based on what she has learned xy 0;egu knx:5and 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 are asked whic8su :motu6g ot;539bqmjdli+ h books they read from a choice of fiction books, non-fiction books and autobiographies. The results are as ft ;5uu6gjdbtqsl8o+ momi39 :ollows.
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 which beverage they prefer fro0swa er/ 1zs r9l1k1s gvmfb92m a choice of coffee, tea and juice. Thr1rk/a9zs v g 210bsfe1slm9we results are 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/ pv70vxte/zlfm*v -iqms/w /tt35cs) for a pizza from Pizzeria da Michele follows a normal distribution with a mean of 30 and standard deviation of 5 . On a particular day, 12 \% of the pizs53l v/cv*twf xmiq-z0v /m7 pset//tzas were delivered in more than 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 supermarkets in a town are suppliedt.e qwvl2s-l))) cpha,2jey j by three major retail suppliers: A, B and C. According to an analysis report, supplier A retains 80 % of their customers each year and lose 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 su,vce-jhyw asp)ltj. )2) e 2qlpplier 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 favoritozr;l t+e)0 zse songs that includes three genres: Jazz, Slow Rock and Country. After her current song ends she ze tzl0ro)s +;randomly selects the next song and the probabilities of genre of the next song are 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 nr)8on kptf 5(y/ 1 xykz0f/gfcew type of probability game to play with her students. In the game, a student draws 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 x5yyo f/t1zkf0 f8r/kpcgn() 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 cpedn sucn,vob 9t5unj;1x378 aught during a fishing cob3,n9n7opuexud5s tvj8 n1; c ntest by 100 contestants.




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 number is composed by the following ru-da2-:m cylr 5t d:lrjles:yra5lj2d d-mc rl: :t-
- 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 pieces and 8 sour pieces. Sarah selects on9fptg - n4w4pk0 sug,ne piece at random and eats it. This piece Sarah selectedfgu-p4 0n9t,gn kwsp4 was sweet. The 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 produap. -57 44zq.kc8clr utunzpw ces LED light strips, 5 \% of which are found to be defectivz u5c-n.84wu tc z7.rppqlka4e.
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 marathon event were asked through -q2)l+ch*cr(w aflic which media channels they received information about tcw*2qha)rl ci+clf - (he 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 information 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 of 10-uv 4;f-bisycro;y* wxa g)elo8m6m;d/ 7qvm0 taxi)ur wqgi a vb;e6dysm-xl7fcm o4 *;;/o- my8v drivers surveyed 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 at the firsi.lshyt5,t d 1t widely known cat show in London at Crystal Palace in 1871. In this early exhibition, oty.d5t1 ih s,lnly 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 nee x phs;1ryprg3*w2;txy.p n/h4ta o9yd 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 S;pa4/3xyx ;.gos rth2wn1 r *yy9hpp ttandard" or "Below Standard". To pass the test, applicants 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 the other foo:nz ,l bt+7iir journals. The probability that a randomly observed visitor accesses the books section is 0.85 , and the probability that the visit tn l:+7,bzoiior accesses the journals section 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 s7 kio7 ona8z:pem 1a0rix students play a soccer passing game. Alex, Bella, Cleo, Dixie, and Emmy stand in a circle and pass the ball to each other while Ben, sae:mo0 71 kz nap7ior8tanding 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 is described by the following transiti bnp q9 xs-+zztmqx;75on 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 experimen8ypi- ,6 afimtt to study the pollination preference of bumblebees' on different floral types. In a flight cage, 240 bumblebees are free to choose between two species of floral: A. majus striatum or A. majus pseudomajus. The changes of pollination behaviors of8tfi a6m-piy , 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 paid travelling vacation to itc)dvx ;cr*ne98b7tbd j, t a6gno c:m,s 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 internation6cogxradtbncm b,nj:d,t * vc;7 9 )e8ally one year change to travel domestically the 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 B, serve in a small city. Each year, a5gh hhl;iwz 2 cj/3qq1u: fi1store A keeps 30 % of its customers while 70 % of them switch to store Store B keeps 60 % of its customers while 40 %iq:fqu5hh1h1lz / iw;j3 c2a g 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 security:wzpz juuxdg;(q0c7. 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 particulb;dh m.; +hhafqm kvzh,x()y+ar species of mammals in two regions; region X and region Y. Each year 30 % of the mammals move from region X to region Y hxa. myhkzbhq+ +f,m( ;h)vd; 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 .