ProbabilityAQA A-Level Maths: Topic test

1. 1
   Events $A$ and $B$ are such that $P(A)=0.3$, $P(B)=0.5$ and $P(A\cup B)=0.65$.

   (a)

   What is the value of $P(A\cap B)$?

   [1 mark]

   • A$0.8$
   • B$0.2$
   • C$0.15$
   • D$0.35$

   (b)

   Which statement about $A$ and $B$ is correct?

   [1 mark]

   • A$A$ and $B$ are independent, because $P(A)\times P(B)=0.15=P(A\cap B)$
   • B$A$ and $B$ are mutually exclusive, because $P(A\cup B)<1$
   • C$A$ and $B$ are not independent, because $P(A\cup B)\ne P(A)+P(B)$
   • D$A$ and $B$ are independent, because $P(A\cap B)=0$

   (c)

   Find $P(A\mid A\cup B)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   At a clinic, 60% of patients are given vaccine $X$ and 40% are given vaccine $Y$. The probability of a mild reaction is $0.1$ for a patient given $X$ and $0.25$ for a patient given $Y$.

   (a)

   What is the probability that a patient chosen at random has a mild reaction?

   [1 mark]

   • A$0.25$
   • B$0.16$
   • C$0.35$
   • D$0.1$

   (b)

   Given that a patient has a mild reaction, what is the probability that they were given vaccine $Y$?

   [1 mark]

   • A$0.4$
   • B$0.25$
   • C$0.16$
   • D$0.625$

   (c)

   Given that a patient does not have a mild reaction, find the probability that they were given vaccine $X$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Events $R$ and $S$ are independent, with $P(R)=0.4$ and $P(S)=p$. It is given that $P(R\cup S)=0.7$.

   (a)

   Find the value of $p$.

   [3 marks]

   (b)

   Find the probability that exactly one of $R$ and $S$ occurs. Show that $R'$ and $S$ are independent.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   In a survey of 100 college students, 55 study Maths ($M$), 40 study Physics ($P$) and 25 study neither subject.

   (a)

   (i) Find the number of students who study both subjects. (ii) Find $P(M\mid P)$. (iii) Determine whether studying Maths and studying Physics are independent for these students.

   [6 marks]

   (b)

   Two students are chosen at random from the 100 students without replacement. (i) Find the probability that both study Physics. (ii) Find the probability that one studies both subjects and the other studies neither. (iii) A teacher says it is acceptable to treat the two selections as independent. Comment, using your answer to (i).

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A delivery company models the probability that a parcel is delivered on time as $0.92$, independently for each parcel. A courier delivers $5$ parcels in one round.

   (a)

   Using the company's model, what is the probability that all $5$ parcels are delivered on time?

   [1 mark]

   • A$0.08$
   • B$0.92$
   • C$0.4$
   • D$0.659$

   (b)

   Which is the best reason why the assumption of independence may be unrealistic?

   [1 mark]

   • AThe parcels have different weights
   • BA traffic jam or breakdown on the round would affect several parcels at once
   • CThe probability $0.92$ is too high to be a probability
   • DThe courier delivers exactly $5$ parcels

   (c)

   Suppose delays tend to affect all the parcels on a round together, but each parcel still has probability $0.92$ of being on time. Explain how this would change the probability that all $5$ parcels are on time compared with $0.659$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   Dev has two coins. Coin 1 is fair. Coin 2 is biased so that the probability of heads is $0.7$. He chooses one of the coins, each being equally likely, and uses that same coin for all his tosses.

   (a)

   What is the probability that a single toss gives heads?

   [1 mark]

   • A$0.6$
   • B$0.35$
   • C$0.7$
   • D$0.5$

   (b)

   Given that the first toss is heads, what is the probability that Dev chose coin 2?

   [1 mark]

   • A$0.35$
   • B$0.5$
   • C$\dfrac{7}{12}$
   • D$0.7$

   (c)

   Dev tosses the chosen coin twice. Find the probability of two heads, and explain why this is not $0.6^2$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A box contains 12 chocolates, of which 7 are milk and 5 are dark. Maya takes 3 chocolates at random, one after the other, without replacement.

   (a)

   Find the probability that all three chocolates are milk.

   [3 marks]

   (b)

   It is given that the probability that exactly one of the three chocolates is dark is $\dfrac{21}{44}$. Find the probability that exactly one is dark, given that at least one is dark.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A hospital screens people for a condition that affects 2% of the population. The test gives a positive result for 95% of people who have the condition and for 4% of people who do not have it.

   (a)

   (i) Find the probability that a person chosen at random tests positive. (ii) Given that a person tests positive, find the probability that they have the condition. (iii) Explain why this probability is low even though the test is accurate for 95% of people with the condition.

   [6 marks]

   (b)

   Three people are screened. They are treated as independent, each with the probability of a positive test found in part (a)(i). (i) Find the probability that at least one tests positive. (ii) State the assumption made, give one reason why it might not be realistic, and say how the probability in (i) would change if the assumption were removed.

   [6 marks]

   Total for question 8: 12 marks

End of questions