Practice Paper 1 (Cambridge International AS Mathematics)

The times, in minutes, that engineers spent dealing with 30 faults at a power station were recorded to the nearest minute. The results are summarised in the table.

Give a reason why a histogram is a suitable diagram to represent these data.

On the grid, draw a histogram to represent these data.

Estimate the proportion of these faults on which the engineers spent longer than three hours.

The percentage marks obtained by 10 students in a recent statistics test are as follows.

38, 41, 19, 33, 22, 0, 27, 19, 10, 99

Find the mode, the range, the mean and the median of these marks.

Give a reason why the median is likely to be more suitable than the mean as a measure of the average mark.

A café owner records the number of customers served between 12 pm and 2 pm on each weekday for three weeks. The results are as follows.

(i) State the mode and find the median of the number of customers served.

(ii) For these data, $\sum x = 851$ and $\sum x^{2} = 48965$. Calculate the mean and the standard deviation of the number of customers served.

State what feature of the data accounts for the mean being greater than the median.

The heights, in cm, of 100 rockhopper penguins are recorded in the following table.

State the number of penguins that are

(i) at least 50 cm tall

(ii) between 45 cm and 55 cm tall.

A rockhopper penguin is chosen at random. Find the probability that the penguin is

(i) at least 50 cm tall

(ii) between 45 cm and 55 cm tall.

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.

Find the number of different arrangements of the 10 letters in the word ENTHUSIASM.

Find the number of different arrangements of the 7 letters in the word BOREDOM in which the consonants (B, D, M and R) and the vowels (E, O and O) alternate.

An arrangement of the 7 letters in the word BOREDOM is chosen at random.

Find the probability that the consonants and vowels alternate. Give your answer as a fraction in its simplest form.

Zara is a gymnast. The probability that she makes a mistake in any given routine is 0.2, independently of other routines. Zara performs 10 routines in a competition. The random variable $X$ is the number of these routines in which Zara makes a mistake.

Find

(i) $\text{E} ( X )$,

(ii) the standard deviation of $X$.

Find the probability that Zara makes a mistake in

(i) none of her routines,

(ii) exactly two of her routines,

(iii) no more than two of her routines.

Given that Zara makes a mistake in at least 2 of her routines, find the probability that she makes a mistake in exactly 3 of her routines.

Find the probability that $X$ is less than one standard deviation away from the mean.

The masses, in grams, of the tomatoes grown in a greenhouse are normally distributed. 0.1% of these tomatoes have a mass greater than 34.451 grams and 19.66% have a mass less than 14.792 grams.

Find the mean and the standard deviation of the masses of these tomatoes.

Leonidas plays a game with a fair six-sided dice. He rolls the dice until either a 6 is obtained or he has rolled the dice four times. The random variable $X$ denotes the number of times the dice is rolled.

Draw up the probability distribution table for $X$.

Find the probability that the dice is rolled

(i) at most 3 times,

(ii) at least 3 times.

Find the mean and the standard deviation of $X$.