Complete the Venn diagram for these students.

Using the given information, complete the following Venn diagram.

Write down $n\left(S\cap \left(M\cup B\right)\right)$.

Write down the value of $x$.

Write down $n\left(B\cap M\cap S^{\prime }\right)$.

Find the probability, in terms of $n$, that the game will end on her second draw.

Hence, find the value of $y$.

fourth draw.

Find the value of $p$;

the student takes either art class or music class, but not both.

Find the value of $q$.

third draw.

Find the number of students in the school that study Mathematics in English.

Find the probability, in terms of $n$, that the game will end on her first draw.

Assuming that rolls of the die are independent, find the probability that Nicky wins the game.

Find $m$.

Write down the value of the median distance in kilometres (km).

Find the probability that the bus journey takes less than $45$ minutes.

Find the probability that Fiona will arrive on time.

This year, Fiona will go to school on $183$ days.

Calculate the number of days Fiona is expected to arrive on time.

Find the probability that the first ball chosen is labelled $\text{A}$.

Find the probability that the first ball chosen is labelled $\text{A}$ or labelled $\text{N}$.

Find $\sigma$.

Find the value of $b$.

Find the probability that both balls chosen are labelled $\text{N}$.

Find the probability that it will take Fiona between $15$ minutes and $30$ minutes to walk to the bus stop.

Find the probability of rolling two or more red faces.

Hayley plays the game when $n$ = 5. She pays $20 to play and can earn money back depending on the number of draws it takes to obtain a blue marble. She earns no money back if she obtains a blue marble on her first draw. Let M be the amount of money that she earns back playing the game. This information is shown in the following table.

Find the value of $k$ so that this is a fair game.

Show that, after a turn, the probability that Ted adds exactly $10 to his winnings is $\frac{1}{3}$.

One of the students who joined the sports club is chosen at random. Find the probability that this student joined both clubs.

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.

A student is chosen at random from the surveyed students.

Find the probability that this student likes kiwi fruit smoothies given that they like mango smoothies.

Determine whether the events $S$ and $M$ are independent.

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.

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

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

Write down the number of students in the group who take art class.

Find the probability of rolling exactly one red face.

Find the value of $a$.

Ted will always have another turn if he expects an increase to his winnings.

Find the least value of $w$ for which Ted should end the game instead of having another turn.

The first $150$ athletes that completed the race won a prize.

Given that an athlete took between $22$ and $m$ minutes to complete the $5 \text{km}$ race, calculate the probability that they won a prize.

Find the probability that this student is taught in Spanish, given that the student studies Biology.

A girl is selected at random. Find the probability that she takes economics but not history.

Find the number of students in the school that are taught in Spanish.

Find the number of students in the school that study both Biology and Mathematics.

Find the number of surveyed students who did not like any of the three flavours.

Find the probability that this student studies Mathematics.

Show that $x=3$.

Find the probability that the second ball chosen is labelled $\text{A}$, given that the first ball chosen was labelled $\text{N}$.

Show that $A=\pi r^2+\frac{1000000}{r}$.

Find the value of this minimum area.

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

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

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

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

Find the probability that this student studies neither Biology nor Mathematics.