Find the value of

(i)     ${\text{A}}$;

(ii)     ${\text{B}}$;

(iii)     ${\text{C}}$.

Calculate the probability that Tomek wears

(i)     shorts and no tie;

(ii)     no tie;

(iii)     shorts given that he is not wearing a tie.

The conference lasts for two days.

Calculate the probability that Tomek wears trousers on both days.

The conference lasts for two days.

Calculate the probability that Tomek wears trousers on one of the days, and shorts on the other day.

Find the probability that this person is not allergic to nuts.

Find the probability that both people chosen are not allergic to nuts.

Copy and complete the tree diagram.

Find the probability that this adult is allergic to nuts and the liquid turns blue.

Find the probability that the liquid turns blue.

Find the probability that the tested adult is allergic to nuts given that the liquid turned blue.

Estimate the number of employees, from this 38, who are allergic to nuts.

Draw a Venn diagram to represent this information. Label A the set that represents The Art Journal readers, B the set that represents The Beartown News readers, and C the set that represents The Currier readers.

Find the percentage of the population that does not read any of the three newspapers.

Find the percentage of the population that reads exactly one newspaper.

Find the percentage of the population that reads The Art Journal or The Beartown News but not The Currier.

A local radio station states that 83 % of the population reads either The Beartown News or The Currier.

Use your Venn diagram to decide whether the statement is true. Justify your answer.

The population of Beartown is 120 000. The local radio station claimed that 34 000 of the town’s citizens read at least two of the local newspapers.

Find the percentage error in this claim.

Draw a Venn diagram to represent this information.

Find the number of families who own no pets.

Find the percentage of families that own exactly one pet.

A family is chosen at random. Find the probability that they own a cat, given that they own a bird.

A student is chosen at random from the group. Find the probability that the student

(i)     studied Mathematics HL;

(ii)    was examined in French;

(iii)   studied Mathematics HL and was examined in French;

(iv)   did not study Mathematics SL and was not examined in English;

(v)    studied Mathematical Studies SL given that the student was examined in Spanish.

Pam believes that the Mathematics course a student chooses is independent of the language in which the student is examined.

Using your answers to parts (a) (i), (ii) and (iii) above, state whether there is any evidence for Pam’s belief. Give a reason for your answer.

Pam decides to test her belief using a Chi-squared test at the $5\%$ level of significance.

(i)     State the null hypothesis for this test.

(ii)    Show that the expected number of Mathematical Studies SL students who took the examination in Spanish is $41.3$, correct to 3 significant figures.

Write down

(i)     the Chi-squared calculated value;

(ii)    the number of degrees of freedom;

(iii)   the Chi-squared critical value.

State, giving a reason, whether there is sufficient evidence at the $5\%$ level of significance that Pam’s belief is correct.

Write down the total number of fish in the pond.

Leanne catches a fish.

Find the probability that she

(i) catches an undersized bream;

(ii) catches either a flathead or an undersized fish or both;

(iii) does not catch an undersized whiting;

(iv) catches a whiting given that the fish was normal.

Leanne notices that on windy days, the probability she catches a fish is 0.1 while on non-windy days the probability she catches a fish is 0.65. The probability that it will be windy on a particular day is 0.3.

Copy and complete the probability tree diagram below.

Leanne notices that on windy days, the probability she catches a fish is 0.1 while on non-windy days the probability she catches a fish is 0.65. The probability that it will be windy on a particular day is 0.3.

Calculate the probability that it is windy and Leanne catches a fish on a particular day.

Leanne notices that on windy days, the probability she catches a fish is 0.1 while on non-windy days the probability she catches a fish is 0.65. The probability that it will be windy on a particular day is 0.3.

Calculate the probability that Leanne catches a fish on a particular day.

Use your answer to part (e) to calculate the probability that Leanne catches a fish on two consecutive days.

Leanne notices that on windy days, the probability she catches a fish is 0.1 while on non-windy days the probability she catches a fish is 0.65. The probability that it will be windy on a particular day is 0.3.

Given that Leanne catches a fish on a particular day, calculate the probability that the day was windy.

Calculate the value of the common ratio.

Calculate the 10^th term of this sequence.

The k^th term is the first term which is greater than 2000. Find the value of k.

Write in words $(q \wedge \neg r) \Rightarrow \neg p$.

Consider the statement “If the number is a multiple of five, and is not even then it will not end in zero”.

Write this statement in symbolic form.

Consider the statement “If the number is a multiple of five, and is not even then it will not end in zero”.

Write the contrapositive of this statement in symbolic form.

Write down the value of a.

Write down the value of b.

Use the tree diagram to find the probability that an employee encountered traffic and was late for work.

Use the tree diagram to find the probability that an employee was late for work.

Use the tree diagram to find the probability that an employee encountered traffic given that they were late for work.

Find the value of x.

Find the value of y.

Find the number of employees who, in the last year, did not travel to work by car, bicycle or public transportation.

Find $n\left( {\left( {C \cup B} \right) \cap P'} \right)$.

Write down in symbolic form the compound proposition

“If $x$ is a factor of 60 then $x$ is a multiple of 5 or $x$ is not a multiple of 4.”

Write down in words the compound proposition $\neg r \wedge (p\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \vee } q)$.

Copy the following truth table and complete the last three columns.

State why the compound proposition $\neg r \wedge (p\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \vee } q)$ is not a logical contradiction.

A row from the truth table from part (c) is given below.

Write down one value of $x$ that satisfies these truth values.

Draw a Venn diagram to represent the given information using sets labelled C, M and P. Complete the diagram to include the number of families represented in each region.

Find the number of families who

(i) went to the circus only;

(ii) went to the museum and the park but not the circus;

(iii) did not go to any of the three places on the weekend.

A family is chosen at random from the group of 40 families. Find the probability that the family went to

(i) the circus;

(ii) two or more places;

(iii) the park or the circus, but not the museum;

(iv) the museum, given that they also went to the circus.

Two families are chosen at random from the group of 40 families.

Find the probability that both families went to the circus.

Draw a Venn diagram to show this information.

There were 25 tourists in the group and every tourist saw at least one of the three types of animal.

Find the number of tourists that saw Cheetah and Rhino but not Leopard.

There were 25 tourists in the group and every tourist saw at least one of the three types of animal.

Calculate the probability that a tourist chosen at random from the group

(i)     saw Leopard;

(ii)     saw only one of the three types of animal;

(iii)     saw only Leopard, given that he saw only one of the three types of animal.

There were 25 tourists in the group and every tourist saw at least one of the three types of animal.

If a tourist chosen at random from the group saw Leopard, find the probability that he also saw Cheetah.

Write the following argument in symbolic form.

“If the land has been purchased and the building permit has been obtained, then the land can be used for residential purposes.”

In your answer booklet, copy and complete a truth table for the argument in part (a).

Begin your truth table as follows.

Use your truth table to determine whether the argument in part (a) is valid.

Give a reason for your decision.

Write down the inverse of the argument in part (a)

(i)     in symbolic form;

(ii)     in words.

Draw a Venn diagram, using sets labelled $E$, $S$ and $A$, to show this information.

Calculate the value of $x$.

Explain, in words, the meaning of $(E \cup S) \cap A'$.

Write down $n\left( {(E \cup S \cup A)'} \right)$.

Find the probability that a woman selected at random from the group had visited Europe.

Find the probability that a woman selected at random from the group had visited Europe, given that she had visited Asia.

Two women from the group are selected at random.

Find the probability that both women selected had visited South America.

Draw a Venn diagram to represent the given information, using sets labelled $B$, $C$ and $H$.

Show that $x = 3$.

Write down the value of $n(B \cap C)$.

Find the probability that this person

(i)     went on at most one trip;

(ii)     went on the coach trip, given that this person also went on both the helicopter trip and the boat trip.

Write down the value of

(i)     $p$ ;

(ii)     $q$ ;

(iii)     $r$.

(i)     Find the probability that Mike reaches Trap B.

(ii)     Find the probability that Mike reaches Trap A.

(iii)     Find the probability that Mike escapes from the maze.

Sonya, a lab assistant, counts the number of paths that lead to traps or escape doors. She believes that the probability that Mike will be trapped is greater than the probability that he will escape.

State whether Sonya is correct. Give a mathematical justification for your conclusion.

During the first trial Mike escapes.

Given that Mike escaped, find the probability that he went directly from S to Escape Door 3.

Write down a formula for $A$, the surface area to be coated.

Express this volume in ${\text{c}}{{\text{m}}^3}$.

Write down, in terms of $r$ and $h$, an equation for the volume of this water container.

Show that $A = \pi {r^2}\frac{{1\,000\,000}}{r}$.

Show that $A = \pi {r^2} + \frac{{1\,000\,000}}{r}$.

Find $\frac{{{\text{d}}A}}{{{\text{d}}r}}$.

Using your answer to part (e), find the value of $r$ which minimizes $A$.

Find the value of this minimum area.

Find the least number of cans of water-resistant material that will coat the area in part (g).

Represent this information on a Venn Diagram.

There were $28$ students who used a bus to travel to school. Calculate the number of students
(i)     who travelled by car and by bus but did not walk;
(ii)    who travelled by car.

Tomoko used a bus to travel to school yesterday.

Find the probability that she also walked.

Two students are chosen at random from all $50$ students.

Find the probability that
(i)     both students walked;
(ii)    only one of the students walked.

Copy the tree diagram below and fill in all the probabilities, where NL represents not late, to represent this information.

Find the probability that Geraldine travels by bus and is late.

Find the probability that Geraldine is late.

Find the probability that Geraldine travelled by train, given that she is late.

Sketch the curve of the function $f (x) = x^3 − 2x^2 + x − 3$ for values of $x$ from −2 to 4, giving the intercepts with both axes.

On the same diagram, sketch the line $y = 7 − 2x$ and find the coordinates of the point of intersection of the line with the curve.

Find the value of the gradient of the curve where $x = 1.7$ .

Draw a Venn diagram to represent the information above.

Write down the number of students who study Music but not Economics.

There are 22 Economics students in total.

(i) Calculate the number of students who study Economics only.

(ii) Find the number of students who study none of these three subjects.

A student is chosen at random from the 100 that were asked above.

Find the probability that this student

(i) studies Economics;

(ii) studies Music and Chemistry but not Economics;

(iii) does not study either Music or Economics;

(iv) does not study Music given that the student does not study Economics.

Jorge applies a ${\chi ^2}$ test at the $5\%$ level to investigate whether wearing a seat belt is associated with the time a driver has had their licence.

(i)     Write down the null hypothesis, ${{\text{H}}_0}$.

(ii)    Write down the number of degrees of freedom.

(iii)   Show that the expected number of drivers that wear a seat belt and have had their driving licence for more than $15$ years is $22$, correct to the nearest whole number.

(iv)   Write down the ${\chi ^2}$ test statistic for this data.

(v)    Does Jorge accept ${{\text{H}}_0}$ ? Give a reason for your answer.

Consider the $200$ drivers surveyed. One driver is chosen at random. Calculate the probability that

(i)     this driver wears a seat belt;

(ii)    the driver does not wear a seat belt, given that the driver has held a licence for more than $15$ years.

Two drivers are chosen at random. Calculate the probability that

(i)     both wear a seat belt.

(ii)    at least one wears a seat belt.

Draw a Venn diagram to show this information. Use T to represent the set of students who have a television, C the set of students who have a computer and I the set of students who have an iPhone.

Write down the number of students that

(i) have a computer only;

(ii) have an iPhone and a computer but no television.

Write down $n[T \cap (C \cup I)']$.

Calculate the number of students who have none of the three.

Two students are chosen at random from the 450 students. Calculate the probability that

(i) neither student has an iPhone;

(ii) only one of the students has an iPhone.

The students are asked to collect money for charity. In the first month, the students collect x dollars and the students collect y dollars in each subsequent month. In the first 6 months, they collect 7650 dollars. This can be represented by the equation x + 5y = 7650.

In the first 10 months they collect 13 050 dollars.

(i) Write down a second equation in x and y to represent this information.

(ii) Write down the value of x and of y .

The students are asked to collect money for charity. In the first month, the students collect x dollars and the students collect y dollars in each subsequent month. In the first 6 months, they collect 7650 dollars. This can be represented by the equation x + 5y = 7650.

In the first 10 months they collect 13 050 dollars.

Calculate the number of months that it will take the students to collect 49 500 dollars.

Find the total number of fish in the net.

Find (i) the modal length interval,

(ii) the interval containing the median length,

(iii) an estimate of the mean length.

(i) Write down an estimate for the standard deviation of the lengths.

(ii) How many fish (if any) have length greater than three standard deviations above the mean?

The fishing company must pay a fine if more than 10% of the catch have lengths less than 40cm.

Do a calculation to decide whether the company is fined.

A sample of 15 of the fish was weighed. The weight, W was plotted against length, L as shown below.

Exactly two of the following statements about the plot could be correct. Identify the two correct statements.

Note: You do not need to enter data in a GDC or to calculate r exactly.

(i) The value of r, the correlation coefficient, is approximately 0.871.

(ii) There is an exact linear relation between W and L.

(iii) The line of regression of W on L has equation W = 0.012L + 0.008 .

(iv) There is negative correlation between the length and weight.

(v) The value of r, the correlation coefficient, is approximately 0.998.

(vi) The line of regression of W on L has equation W = 63.5L + 16.5.

Write down the value of a.

Write down the value of b.

Find the probability that both biscuits are plain.

Copy and complete the tree diagram below.

Find the probability that he chooses a chocolate biscuit.

Find the probability that he chooses a biscuit from the red tin given that it is a chocolate biscuit.

Use the table to find the probability that

(i) a television will be bought by a female;

(ii) a television with a screen size of 32 < S ≤ 46 will be bought;

(iii) a television with a screen size of 32 < S ≤ 46 will be bought by a female;

(iv) a television with a screen size greater than 46 inches will be bought, given that it is bought by a male.

The manager of the store wants to determine whether the screen size is independent of gender. A Chi-squared test is performed at the 1 % significance level.

Write down the null hypothesis.

The manager of the store wants to determine whether the screen size is independent of gender. A Chi-squared test is performed at the 1 % significance level.

Show that the expected frequency for females who bought a screen size of 32 < S ≤ 46, is 79, correct to the nearest integer.

The manager of the store wants to determine whether the screen size is independent of gender. A Chi-squared test is performed at the 1 % significance level.

Write down the number of degrees of freedom.

The manager of the store wants to determine whether the screen size is independent of gender. A Chi-squared test is performed at the 1 % significance level.

Write down the ${\chi ^2}$ calculated value.

The manager of the store wants to determine whether the screen size is independent of gender. A Chi-squared test is performed at the 1 % significance level.

Write down the critical value for this test.

The manager of the store wants to determine whether the screen size is independent of gender. A Chi-squared test is performed at the 1 % significance level.

Determine if the null hypothesis should be accepted. Give a reason for your answer.

Represent this information on a Venn diagram.

Calculate the number of students who own none of the items mentioned above.

If a student is chosen at random, write down the probability that the student owns a digital camera only.

If two students are chosen at random, calculate the probability that they both own a cell phone only.

If a student owns an iPod, write down the probability that the student also owns a digital camera.

Copy and complete the probability tree diagram below.

Calculate the probability that

(i)     Kate goes to the cinema and is not late;

(ii)    the person who goes to the cinema arrives home late.

Draw a Venn Diagram, using sets labelled $A$ , $S$ and $P$ , that shows this information.

There were 48 customers of Pete’s Eats that day. Calculate the number of people who were customers of all three cafés.

There were 50 customers of Sarah’s Snackbar that day. Calculate the total number of people who were customers of Alan’s Diner.

Write down the number of customers of Alan’s Diner that were also customers of Pete’s Eats.

Find $n[(S \cup P) \cap A']$.

One of the survey forms was chosen at random, find the probability that the form showed “Dissatisfied”;

One of the survey forms was chosen at random, find the probability that the form showed “Satisfied” and was completed at Sarah’s Snackbar;

One of the survey forms was chosen at random, find the probability that the form showed “Dissatisfied”, given that it was completed at Alan’s Diner.

A ${\chi ^2}$ test at the $5\%$ significance level was carried out to determine whether there was any difference in the level of customer satisfaction in each of the cafés.

Write down the null hypothesis, ${{\text{H}}_0}$ , for the ${\chi ^2}$ test.

A ${\chi ^2}$ test at the $5\%$ significance level was carried out to determine whether there was any difference in the level of customer satisfaction in each of the cafés.

Write down the number of degrees of freedom for the test.

A ${\chi ^2}$ test at the $5\%$ significance level was carried out to determine whether there was any difference in the level of customer satisfaction in each of the cafés.

Using your graphic display calculator, find ${\chi ^2}_{calc}$ .

A ${\chi ^2}$ test at the $5\%$ significance level was carried out to determine whether there was any difference in the level of customer satisfaction in each of the cafés.

State, giving a reason, the conclusion to the test.

Copy and complete the Venn diagram on your answer paper.

Write down the number of students who study Mathematics but not Further Mathematics.

Write down the total number of students in the school.

Write down $n({\text{B}} \cup {\text{C}})$.

Write the following compound statement in symbolic form.

“It is snowing and the roads are not open.”

Write the following compound statement in words.

$(\neg p \wedge q) \Rightarrow r$

An incomplete truth table for the compound proposition $(\neg p \wedge q) \Rightarrow r$ is given below.

Copy and complete the truth table on your answer paper.

Represent this information on a Venn diagram.

Find the number of students who had none of the three choices for breakfast.

Write down the percentage of students who had fruit for breakfast.

Describe in words what the students in the set $X \cap Y'$ had for breakfast.

Find the probability that a student had at least two of the three choices for breakfast.

Two students are chosen at random. Find the probability that both students had all three choices for breakfast.

Write down the null hypothesis, H₀, for this test.

Write down the number of degrees of freedom for this test.

Write down the critical value for this test.

Show that the expected number of females that have more than 5 meals per day is 13, correct to the nearest integer.

Use your graphic display calculator to find the $\chi _{calc}^2$ for this data.

Decide whether H₀ must be accepted. Justify your answer.

Copy and complete the tree diagram to represent this information.

Find the probability that Lisa cooks dinner and they do not have pasta.

Find the probability that they do not have pasta.

Given that they do not have pasta, find the probability that Lisa cooked dinner.

Write down $n(R \cap S \cap B)$.

Find $n(R')$.

Describe which groups the pupils in the set $S \cap B$ like.

Use set notation to describe the group of pupils who like the Rockers and the Bluers but do not like the Salseros.

There are 33 pupils in the class.

Find x.

There are 33 pupils in the class.

Find the number of pupils who like the Rockers.

Write down the null hypothesis, H₀, for this test.

Find the expected value of female footballers.

Write down the number of degrees of freedom.

Write down the critical value of $\chi ^2$, at the 5 % level of significance.

Use your graphic display calculator to determine the $\chi _{calc}^2$ value.

Determine whether H₀ should be accepted. Justify your answer.

One student is chosen at random from the 120 students.

Find the probability that this student

(i) is male;

(ii) plays tennis.

Two students are chosen at random from the 120 students.

Find the probability that

(i) both play football;

(ii) neither play basketball.

Copy the following Venn diagram and correctly insert all values from the above information.

The number of customers that like only mangoes is equal to the number of customers that like only kiwi fruits. This number is half of the number of customers that like only bananas.

Complete your Venn diagram from part (a) with this additional information in terms of $x$.

The number of customers that like only mangoes is equal to the number of customers that like only kiwi fruits. This number is half of the number of customers that like only bananas.

Find the value of $x$.

The number of customers that like only mangoes is equal to the number of customers that like only kiwi fruits. This number is half of the number of customers that like only bananas.

Write down the number of customers who like

(i)     mangoes;

(ii)     mangoes or bananas.

The number of customers that like only mangoes is equal to the number of customers that like only kiwi fruits. This number is half of the number of customers that like only bananas.

A customer is chosen at random from the 100 customers. Find the probability that this customer

(i)     likes none of the three fruits;

(ii)     likes only two of the fruits;

(iii)     likes all three fruits given that the customer likes mangoes and bananas.

The number of customers that like only mangoes is equal to the number of customers that like only kiwi fruits. This number is half of the number of customers that like only bananas.

Two customers are chosen at random from the 100 customers. Find the probability that the two customers like none of the three fruits.

One of the dogs is chosen at random.

(i) Find P (the dog is grey and has the yellow bowl).

(ii) Find P (the dog does not get the yellow bowl).

Neil often takes the dogs to the park after they have eaten. He has noticed that the grey dog plays with a stick for a quarter of the time and both brown dogs play with sticks for half of the time. This information is shown on the tree diagram below.

(i) Copy the tree diagram and add the four missing probability values on the branches that refer to playing with a stick.

During a trip to the park, one of the dogs is chosen at random.

(ii) Find P (the dog is grey or is playing with a stick, but not both).

(iii) Find P (the dog is grey given that the dog is playing with a stick).

(iv) Find P (the dog is grey and was fed from the yellow bowl and is not playing with a stick).

Complete the diagram, using the information given in the question.

Find (i) $n(M \cap W)$

(ii) $n(M′ \cup S)$

Two mice are chosen without replacement.

Find P (both mice are short-tailed).

State the alternative hypothesis.

Calculate the expected frequency of flights travelling at most 500 km and arriving slightly delayed.

Write down the number of degrees of freedom.

Write down the χ² statistic.

Write down the associated p-value.

State, with a reason, whether you would reject the null hypothesis.

Write down the probability that this flight arrived on time.

Given that this flight was not heavily delayed, find the probability that it travelled between 500 km and 5000 km.

Two flights are chosen at random from those which were slightly delayed.

Find the probability that each of these flights travelled at least 5000 km.

One person is chosen at random from those surveyed. Find the probability that this person

(i) does not prefer Zucos;

(ii) prefers Chocos, given that they live in Montevideo.

Two people are chosen at random from those surveyed. Find the probability that they both prefer Fruti.

The market research organization tests the survey data to determine whether the brand of cereal preferred is associated with a city. A chi-squared test at the $5\%$ level of significance is performed.

State the null hypothesis.

The market research organization tests the survey data to determine whether the brand of cereal preferred is associated with a city. A chi-squared test at the $5\%$ level of significance is performed.

State the number of degrees of freedom.

The market research organization tests the survey data to determine whether the brand of cereal preferred is associated with a city. A chi-squared test at the $5\%$ level of significance is performed.

Show that the expected frequency for the number of people who live in Montevideo and prefer Zucos is $63$.

The market research organization tests the survey data to determine whether the brand of cereal preferred is associated with a city. A chi-squared test at the $5\%$ level of significance is performed.

Write down the chi-squared statistic for this data.

The market research organization tests the survey data to determine whether the brand of cereal preferred is associated with a city. A chi-squared test at the $5\%$ level of significance is performed.

State whether the market research organization would accept the null hypothesis. Clearly justify your answer.

Plot these pairs of values on a scatter diagram. Use a scale of $1{\text{ cm}}$ to represent $50$ pages on the horizontal axis and $1{\text{ cm}}$ to represent $1{\text{ AUD}}$ on the vertical axis.

Write down the linear correlation coefficient, $r$, for the data.

Stephen wishes to buy a paperback book which has $350$ pages in it. He plans to draw a line of best fit to determine the price. State whether or not this is an appropriate method in this case and justify your answer.

$180$ people were interviewed and asked what types of transport they had used in the last year from a choice of airplane $(A)$, train $(T)$ or bus $(B)$. The following information was obtained.

$47$ had travelled by airplane

$68$ had travelled by train

$122$ had travelled by bus

$25$ had travelled by airplane and train

$32$ had travelled by airplane and bus

$35$ had travelled by train and bus

$20$ had travelled by all three types of transport

Draw a Venn diagram to show the above information.

Find the number of people who, in the last year, had travelled by

(i)      bus only;

(ii)     both airplane and bus but not by train;

(iii)    at least two types of transport;

(iv)    none of the three types of transport.

A person is selected at random from those who were interviewed.

Find the probability that the person had used only one type of transport in the last year.

Given that the person had used only one type of transport in the last year, find the probability that the person had travelled by airplane.

Write down the probability that Ayako avoids the trap in this wall.

Find the probability that only one of Ayako and Natsuko falls into a trap while attempting to pass through a door in the first wall.

Copy the probability tree diagram and write down the relevant probabilities along the branches.

A contestant is chosen at random. Find the probability that this contestant fell into a trap while attempting to pass through a door in the second wall.

A contestant is chosen at random. Find the probability that this contestant fell into a trap.

120 contestants attempted this game.

Find the expected number of contestants who fell into a trap while attempting to pass through a door in the third wall.

A group of students at Dune Canyon High School were surveyed. They were asked which of the following products: books (B), music (M) or films (F), they downloaded from the internet.

The following results were obtained:

100 students downloaded music;
95 students downloaded films;
68 students downloaded films and music;
52 students downloaded books and music;
50 students downloaded films and books;
40 students downloaded all three products;
8 students downloaded books only;
25 students downloaded none of the three products.

Use the above information to complete a Venn diagram.

Calculate the number of students who were surveyed.

i)     On your Venn diagram, shade the set ${\left( {F \cup M} \right) \cap B'}$ . Do not shade any labels or values on the diagram.

ii)    Find $n\left( {\left( {F \cup M} \right) \cap B'} \right)$ .

A student who was surveyed is chosen at random.

Find the probability that

(i)     the student downloaded music;

(ii)    the student downloaded books, given that they had not downloaded films;

(iii)   the student downloaded at least two of the products.

Dune Canyon High School has 850 students.

Find the expected number of students at Dune Canyon High School that downloaded music.