Practice Paper 2 (Cambridge International AS Mathematics)

A charity shows an advert on TV at the same time every weekday for four weeks. The manager records the number of donations received in the hour after the advert each day. The results are as follows.

Represent these data in a stem-and-leaf diagram.

The manager decides that the advert is only worth continuing if the median number of donations in the hour after the advert is at least 25.

Determine whether the manager should continue to run the advert.

Give one advantage of using a stem-and-leaf diagram, rather than a grouped frequency table, to represent these data.

A set of values, $x$, is given below.

12, 10, 8, 6, 13, 11, 9, 12

(i) Create a new set of values, $d$, by subtracting 10 from each value of $x$.

(ii) Find the mean of the values of $d$.

(iii) Show that the mean of the values of $d$ is 10 less than the mean of the values of $x$.

Joe counts the number of different species of bird visiting his garden each day for a week. The results are given below.

Calculate the mean number of different species of bird visiting Joe's garden.

Joe continues to record the number of different species of bird visiting his garden each day for the rest of the month and calculates the mean number of different species is 9.25 for the remaining 24 days.

Joe says, using the data from the whole month, he would expect to see 9 different species of bird every day. Explain whether Joe is correct. You must support your answer with clear working.

Joe decides he will repeat his investigation into species of bird visiting his garden three months later. That month he finds the mean number of different species visiting his garden per day is 4.5. Joe is concerned that this indicates some species of bird are dying out. Suggest a reason why this may not be the case.

Two letters are chosen at random, without replacement, from the 10 letters of the word IMPOSSIBLE.

Event $A$ is 'the two letters are the same'.

Event $B$ is 'the first letter chosen is an S'.

Find the probability that

(i) $A$ occurs

(ii) $B$ occurs

(iii) both $A$ and $B$ occur.

Determine whether events $A$ and $B$ are mutually exclusive, justifying your answer.

In an online chess tournament, each match begins with a standard game, which is either won outright, lost outright or drawn. If the standard game is drawn, a single sudden-death game is played to decide the winner of the match. A sudden-death game cannot be drawn. A player who wins a match goes on to play another match.

In his first match, the probability that Jamie wins outright is 0.7 and the probability that he loses outright is 0.1.

If Jamie wins his first match outright, the probabilities that he wins or loses his second match outright are again 0.7 and 0.1.

If Jamie wins his first match after a sudden-death game, the probability that he wins his second match outright is 0.55 and the probability that he loses it outright is 0.3.

The probability that Jamie wins any sudden-death game is $p$.

Given that the probability that Jamie wins both of his first two matches is 0.6508, find the value of $p$.

Find the probability that Jamie won both of his first two matches outright, given that he won both of these matches.

State whether each of the following scenarios represents a permutation or a combination.

(i) A pin code is a 4-digit number made up from the digits 0 to 9, using each digit once only.

(ii) Three students from a class of twelve are chosen at random to represent their class in an interview.

(iii) Five toppings are chosen from a menu of twelve different toppings to put on a pizza.

(iv) A student council is made up of ten members and they need to elect a president, a vice-president and a treasurer.

Bars of a particular brand of chocolate are labelled as weighing 300 g. The probability that a randomly chosen bar weighs between 297 g and 303 g is 0.9596. For bars outside this range, the probability that a bar weighs less than 297 g is equal to the probability that it weighs more than 303 g.

Millie buys 25 of these bars, which may be regarded as a random sample. The random variable $U$ is the number of these bars that weigh less than 297 g.

State the distribution of $U$, giving the values of its parameters.

Millie plans to hand out the bars at a meeting attended by 24 people. Bars weighing less than 297 g cannot be handed out.

Find the probability that there will be enough bars to hand out to

(i) all 24 people,

(ii) all but one of the 24 people, but not all of them.

The heights, in metres, of the trees on a plantation are normally distributed with mean $μ$ and standard deviation $σ$. The shortest 10% of the trees and the tallest 2.5% of the trees are not sold. All the other trees are sold.

Find, in terms of $μ$ and $σ$, the least height and the greatest height of the trees that are sold.

It is given that 20% of the trees are shorter than $( μ - 8 )$ m.

Find the probability that a randomly chosen tree is taller than $( μ + 8 )$ m, given that it is taller than $( μ - 8 )$ m.

The random variable $X$ takes the values 1, 3, 5 and 7 only. The probability that $X$ takes the value $x$ is $k x$, where $k$ is a constant.

Find the value of $k$.

Find $\text{P} ( X > 3 )$.

For each of the following, state with a reason whether the random variable in question is a discrete random variable or a continuous random variable.

(i) 100 red squirrels from the wild are sampled. The random variable $A$ is the tail length of a randomly selected red squirrel.

(ii) 100 students sit a test which is marked out of 50. The random variable $B$ is the number of marks achieved by a randomly selected student.

(iii) 100 men are in a shoe shop. The random variable $C$ is the shoe size of a randomly selected man.

The following histogram shows the distribution of results when a large number of measurements of the specified random variable $D$ are made.

State with a reason whether a normal distribution would be appropriate for modelling the random variable.

A company tests elastic bands by stretching them until they snap. The length, in millimetres, of each band at the moment it snaps is recorded.

The incomplete histogram and frequency table below show the results.

Use the information to complete both the histogram and the frequency table.

The numbers of goals scored by the 24 teams in a football tournament are summarised in the table below.

Calculate an estimate of the mean number of goals scored per team.

Calculate an estimate of the standard deviation of the number of goals scored per team.

The ages, $a$ years, of the 27 children at a birthday party are summarised in the table. No child at the party was older than 7 years.

Calculate estimates of the mean and the standard deviation of the ages of the children.

State one disadvantage of grouping data before analysing it.

Two more children, both aged 3 years, arrive at the party.

Without doing any further calculations, state with a reason whether each of your estimates in part (a) would increase, decrease or stay the same.

Isabel has two boxes of chocolates. Box $A$ contains 4 caramels and 6 toffees, and box $B$ contains 5 caramels and $x$ toffees. Isabel chooses a chocolate at random from box $A$ and places it in box $B$. Her friend then chooses a chocolate at random from box $B$.

Show that the probability that Isabel's friend chooses a caramel is $\frac{27}{5(x+6)}$.

Given that the probability that Isabel's friend chooses a toffee is $\frac{11}{20}$, find the value of $x$.

A basket contains 6 green apples and 8 red apples. Five apples are chosen at random from the basket, without replacement.

(i) Find the probability that 2 red apples and 3 green apples are chosen.

(ii) Given that the first 4 apples chosen are green, find the probability that the fifth apple chosen is green.

A second basket contains 4 green apples and 5 red apples. The apples in the two baskets are combined, and 2 apples are chosen at random from the combined apples, without replacement.

Find the probability that the 2 apples are from the same basket, given that they are the same colour.