Basic Probability (AQA A Level Maths: Statistics): Exam Questions

$A$, $B$, $C$ and $D$ are four events such that $\text{P}\left(A\right)=0.28$, $\text{P}\left(B\right)=0.33$, $\text{P}\left(C\right)=0.43$ and $\text{P}\left(D\right)=0.25$.

(i) Given that events $B$ and $C$ are mutually exclusive, find $\text{P}\left(B\text{ or }C\right)$.

(ii) Given that events $A$ and $D$ are independent, find $\text{P}\left(A\text{ and }D\right)$.

A burger bar collects data on whether customers order two of its signature items: a burger and a portion of fries. The probability that a customer orders both a burger and fries is $0.12$. The probability that a customer orders only one of the two items is $0.71$. The probability that a customer orders fries but does not order a burger is $0.03$.

Draw a Venn diagram to represent this information.

Determine, giving a reason for your answer, whether the events ‘orders a burger’ and ‘orders fries’ are independent.

A college surveys a group of Year 12 students about which of three sports they regularly play. The Venn diagram below shows the probabilities that a randomly selected student plays each sport.

$A$ represents the event that the student plays tennis.

$B$ represents the event that the student plays badminton.

$C$ represents the event that the student plays squash.

Given that the probability that a student plays squash is $0.44$, find the values of

(i) $x$

(ii) $y$.

Find the probability that a randomly selected student

(i) plays at least one of the three sports

(ii) plays exactly one of the three sports.

On any given day the probability that Tomás reads poetry is $0.55$ and the probability that he visits the pub is $0.75$. On every day, he does at least one of these two activities.

Find the probability that on a given day Tomás

(i) both reads poetry and visits the pub

(ii) visits the pub but does not read poetry.

$A$ and $B$ are two events such that $\text{P}\left(A\right)=0.4$, $\text{P}\left(B\right)=0.8$ and $\text{P}\left(A\text{ and }B\right)=0.35$.

Find

(i) the probability that event $B$ occurs but event $A$ does not occur

(ii) the probability that neither event $A$ nor event $B$ occurs.

An ice cream company surveys a group of customers to find out whether they like the company’s two new flavours: chocolate and strawberry. The probability that a customer likes chocolate is $0.2$. The probability that a customer likes strawberry is $0.15$. The probability that a customer likes neither flavour is $0.68$.

Draw a Venn diagram to represent this information.

Determine, giving a reason for your answer, whether the events ‘a customer likes chocolate’ and ‘a customer likes strawberry’ are independent.

A committee is made up of $13$ students and $7$ teachers. Two members are selected at random, without replacement, to represent the committee at a meeting.

Draw a tree diagram to represent this selection.

Find the probability that the two members selected are of the same type (i.e. both students or both teachers).

A manufacturer tests components using a two-stage quality control process.

• At stage 1 the probability that a component passes is $0.7$.

• If a component passes stage 1, there is a probability of $0.8$ that it is accepted for sale, otherwise it is sent for rework.

• If a component fails stage 1, there is a probability of $0.2$ that it is sent for rework, otherwise it is scrapped.

Draw a tree diagram to represent this information.

Find the probability that a randomly chosen component is

(i) accepted for sale

(ii) sent for rework

(iii) scrapped.

Determine, giving a reason for your answer, whether the events ‘passes stage 1’ and ‘sent for rework’ are independent.

A game is played using a fair spinner with four sectors numbered $1$ to $4$ and a fair dice with its six sides numbered $1$ to $6$. The spinner is spun and the dice is rolled, and the score in the game is the positive difference between the two results.

Find the probability that the score in the game is

(i) exactly $0$

(ii) $3$ or more

(iii) a prime number.

The histogram below shows the distribution of masses, in grams, of 80 newly hatched ducklings:

Find the probability that a duckling chosen at random has a mass less than 54 g.

Estimate the probability that a duckling chosen at random has a mass greater than 53 g.

A college surveys a group of Year 12 students about which of three sciences they study. The Venn diagram below shows the probabilities that a randomly chosen student studies each subject.

$A$ represents the event that the student studies biology.

$B$ represents the event that the student studies chemistry.

$C$ represents the event that the student studies physics.

The probability that a student studies at least one of the three sciences is $0.92$ and the probability that a student studies exactly one of the three sciences is $0.7$.

Find the values of $x$, $y$ and $z$.

Find the probability that a randomly selected student

(i) studies chemistry

(ii) studies exactly two of the three sciences.

A shipment of $20$ components contains $12$ that are flawless, $7$ with a minor defect and $1$ with a major defect. Two components are selected at random, without replacement, for inspection.

Draw a tree diagram to represent this selection.

Find the probability that the two components selected are not both of the same type.

A customer reports a fault with a product. A technician will attempt up to three times to fix it.

• On the first attempt, the probability the fix is successful is $0.7$.

• If the first attempt fails, a second attempt is made, with probability $0.6$ of success.

• If the second attempt also fails, a third attempt is made, with probability $0.5$ of success.

Each time a fix is successful, there is a probability of $0.2$ that the customer accepts a free upgrade (as compensation) and a probability of $0.8$ that the fault is simply resolved with no upgrade.

If all three attempts fail, the product is replaced.

Draw a tree diagram to represent this information.

Find the probability that

(i) the customer accepts a free upgrade

(ii) the fault is resolved with no upgrade

(iii) the product is replaced.

The histogram below shows the distribution of masses, in kg, of $100$ newborn calves:

Find the probability that a calf chosen at random has a mass greater than $21$ kg.

Estimate the probability that a calf chosen at random has a mass less than 23.5 kg.

$A$, $B$ and $C$ are three events such that $\text{P}\left(A\right)=0.28$, $\text{P}\left(A\text{ or }C\right)=0.4$ and $\text{P}\left(B\text{ and }C\right)=0.02$.

Given that $A$ and $C$ are mutually exclusive, and that $B$ and $C$ are independent, find

(i) $\text{P}\left(C\right)$

(ii) $\text{P}\left(B\right)$

(iii) the probability that event $C$ occurs but event $B$ does not occur.

The Venn diagram below shows the probabilities of members of a horror film society having seen various films.

$A$ represents the event that the member has seen Aaaaaaaaaaagh!.

$B$ represents the event that the member has seen Beware the Gloaming.

$C$ represents the event that the member has seen Cute Kittens of Doom.

Given that half the members of the society have seen Cute Kittens of Doom, and that $38\%$ of members have seen at least two of the three films, find the values of $x$, $y$ and $z$.

Find the probability that a randomly selected member of the society

(i) has seen exactly two of the three films

(ii) has seen at least one of the three films but not all three of them.

A game is played using a fair spinner with four sectors numbered $1$ to $4$ and a fair eight-sided dice with its sides numbered $1$ to $8$. The spinner is spun and the dice is rolled, and the score in the game is determined as follows:

• if the number on the spinner is higher than the number on the dice, the score is the sum of the two numbers

• if the number on the spinner is lower than the number on the dice, the score is the positive difference of the two numbers

• if the numbers on the spinner and the dice are equal, the score is the product of the two numbers.

Find the probability that the score in the game is

(i) exactly $7$

(ii) $10$ or more

(iii) a triangular number (i.e. one of $1,3,6,10,15,21,\dots$).

On any given day the probability that Björn has an avocado with his lunch is $0.39$, while the probabilities that he has a bacon butty or a piece of carrot cake are $0.32$ and $0.44$ respectively. He never has all three items on the same day, but the probability that he has at least one of them is $0.91$.

The probability that he has exactly two of the three items is the same regardless of which two items those are.

Find the probability that on a given day Björn has

(i) exactly one of the three items

(ii) a bacon butty but not a piece of carrot cake.

A music streaming service collects data about the types of music listened to by its users. The probability that a user listens to pop is $0.25$. The probability that a user listens to rock is $0.36$. The probability that a user listens to classical is $0.56$.

Only $10\%$ of users listen to none of the three types of music. No user listens to both pop and classical. The probability that a user listens to both rock and classical is twice the probability that a user listens to both pop and rock.

Draw a Venn diagram to represent this information.

Determine whether any of the events ‘listens to pop’, ‘listens to rock’ and ‘listens to classical’ are independent of one another.

A bag contains $10$ black tokens and $6$ white tokens. A token is drawn from the bag at random and its colour recorded, and then a fair coin is flipped.

If the coin lands on heads, a second token is drawn from the bag without replacing the first token.

If the coin lands on tails, the first token is replaced in the bag before a second token is drawn.

Draw a tree diagram to represent this experiment.

Find the probability that the second token drawn is white.