Statistics: ProbabilityEdexcel A-Level Maths: Topic test

1. 1
   Events $R$ and $S$ are independent, with $P(R)=0.4$ and $P(S)=0.3$.

   (a)

   What is the value of $P(R\cap S)$?

   [1 mark]

   • A$0.7$
   • B$0.12$
   • C$0$
   • D$0.1$

   (b)

   What is the value of $P(R\mid S)$?

   [1 mark]

   • A$0.3$
   • B$0.12$
   • C$0.7$
   • D$0.4$

   (c)

   Find $P(R'\cap S')$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A survey of 120 adults records whether each person cycles to work and whether each person owns a car. 45 adults cycle to work, 70 own a car and 20 adults do both.

   (a)

   How many of the 120 adults neither cycle to work nor own a car?

   [1 mark]

   • A$25$
   • B$5$
   • C$45$
   • D$50$

   (b)

   What is the probability that a randomly chosen adult cycles to work, given that they own a car?

   [1 mark]

   • A$\dfrac{20}{120}$
   • B$\dfrac{20}{45}$
   • C$\dfrac{2}{7}$
   • D$\dfrac{45}{120}$

   (c)

   Find the probability that a randomly chosen adult owns a car, given that they do not cycle to work.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A commuter's route has a traffic light. A model assumes that on each morning the light is red when the commuter arrives with probability $0.35$, independently of every other morning.

   (a)

   Find the probability that the light is red on exactly one of the next three mornings.

   [3 marks]

   (b)

   In fact the light is more likely to be red on a morning when it was red the day before. Suppose $P(\text{red on a morning}\mid\text{red the previous morning})=0.6$. Compare $P(\text{red on two consecutive mornings})$ under this suggestion with the value from the original model, and comment on the original model.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   At a college, $60\%$ of students study at least one science subject. Of the students who study at least one science subject, $30\%$ are in a sports team. Of the students who study no science subject, $50\%$ are in a sports team. Let $S$ be the event that a randomly chosen student studies at least one science subject and $T$ the event that the student is in a sports team.

   (a)

   Find $P(T)$ and hence find the probability that a randomly chosen student who is in a sports team studies at least one science subject.

   [6 marks]

   (b)

   Determine whether $S$ and $T$ are independent and whether they are mutually exclusive. Two students are chosen at random. Assuming that their choices are independent, find the probability that exactly one of them is in a sports team, and state one reason why the assumption of independence is reasonable.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   Events $E$ and $F$ are mutually exclusive, with $P(E)=0.15$ and $P(F)=0.45$.

   (a)

   What is the value of $P(E\cup F)$?

   [1 mark]

   • A$0.0675$
   • B$0.3$
   • C$0.6$
   • D$0.5325$

   (b)

   Which statement about $E$ and $F$ is correct?

   [1 mark]

   • A$E$ and $F$ are not independent.
   • B$E$ and $F$ are independent, because $P(E\cap F)=0$.
   • C$E$ and $F$ are independent, because they cannot both occur.
   • D$P(E\mid F)=P(E)$.

   (c)

   Event $G$ is independent of $E$, with $P(G)=0.5$. Find $P(E\cup G)$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   For two events $X$ and $Y$, $P(X)=0.55$, $P(Y)=0.4$ and $P(X\cup Y)=0.7$.

   (a)

   What is the value of $P(X\cap Y)$?

   [1 mark]

   • A$0.95$
   • B$0.3$
   • C$0.22$
   • D$0.25$

   (b)

   What is the value of $P(X\mid Y)$?

   [1 mark]

   • A$0.25$
   • B$0.625$
   • C$0.4545$
   • D$0.55$

   (c)

   Find $P(Y\mid X')$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A cafe models each customer's choice of drink as independent of every other customer, with probabilities $0.55$ for coffee, $0.30$ for tea and $0.15$ for another drink.

   (a)

   Find the probability that two customers chosen at random order different types of drink.

   [3 marks]

   (b)

   The cafe records 200 pairs of customers who arrived together, and in 110 of these pairs both customers ordered the same type of drink. Use the model to find the expected number of such pairs and comment on whether the model is appropriate for customers who arrive together.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A factory has two machines. Machine $P$ makes $60\%$ of the items and machine $Q$ makes $40\%$. Of the items made by $P$, $3\%$ are defective. Of the items made by $Q$, $5\%$ are defective.

   (a)

   An item is chosen at random from the factory's output. Find the probability that it is defective. Given that it is defective, find the probability that it was made by $Q$ and comment on this value compared with the share of items that $Q$ makes.

   [6 marks]

   (b)

   Two items are chosen at random from a day's output. Assuming that whether they are defective is independent, find the probability that both are defective and the probability that at least one is defective. Give one reason why the assumption of independence is reasonable and one reason why it may not be.

   [6 marks]

   Total for question 8: 12 marks

End of questions