S1: ProbabilityEdexcel International A Level Maths: Topic test

1. 1
   Events $R$ and $S$ are such that $P(R)=0.55$, $P(S)=0.30$ and $P(R\cup S)=0.70$.

   (a)

   Find $P(R\cap S)$.

   [1 mark]

   • A$0.85$
   • B$0.40$
   • C$0.165$
   • D$0.15$

   (b)

   Which of the following statements is correct?

   [1 mark]

   • A$R$ and $S$ are mutually exclusive.
   • B$R$ and $S$ are independent.
   • C$R$ and $S$ are neither mutually exclusive nor independent.
   • D$R$ and $S$ are both mutually exclusive and independent.

   (c)

   Find $P(R\mid S)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A club has 50 members. Of these, 32 swim, 24 cycle and 10 do neither. One member is chosen at random. Let $S$ be the event that the member swims and $C$ the event that the member cycles.

   (a)

   How many members do both?

   [1 mark]

   • A16
   • B6
   • C40
   • D22

   (b)

   Find $P(S\mid C)$.

   [1 mark]

   • A$\frac12$
   • B$\frac23$
   • C$\frac{8}{25}$
   • D$\frac25$

   (c)

   Determine whether $S$ and $C$ are independent, showing your working.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A box contains 4 red pens, 3 green pens and 5 yellow pens. Two pens are taken at random from the box without replacement.

   (a)

   Find the probability that the two pens are the same colour.

   [3 marks]

   (b)

   Given that at least one of the two pens is yellow, find the probability that both pens are yellow.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A year group has 100 students. Of these, 48 study biology ($B$), 40 study chemistry ($C$) and 35 study physics ($P$). Also, 15 study both $B$ and $C$, 12 study both $B$ and $P$, 10 study both $C$ and $P$, and 5 study all three subjects.

   (a)

   One student is chosen at random.
   (i) Find the probability that the student studies none of the three subjects. [2]
   (ii) Find the probability that the student studies exactly two of the three subjects. [2]
   (iii) Given that the student studies exactly one of the three subjects, find the probability that the student studies biology. [2]

   [6 marks]

   (b)

   Two different students are chosen at random from the year group.
   (i) Find the probability that both study biology. [2]
   (ii) Find the probability that exactly one of them studies physics. [2]
   (iii) Show that the events 'a randomly chosen student studies biology' and 'a randomly chosen student studies chemistry' are not independent. [2]

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A card is chosen at random from 20 cards numbered 1 to 20. Let $M$ be the event that the number is a multiple of 3, and $E$ the event that it is even.

   (a)

   Find $P(M\cup E)$.

   [1 mark]

   • A$\frac{4}{5}$
   • B$\frac{3}{20}$
   • C$\frac{13}{20}$
   • D$\frac{7}{20}$

   (b)

   Find $P(M\mid E)$.

   [1 mark]

   • A$\frac{3}{10}$
   • B$\frac12$
   • C$\frac{3}{20}$
   • D$\frac35$

   (c)

   Determine whether $M$ and $E$ are independent.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A biased coin has probability 0.6 of landing heads on each toss, independently of every other toss. The coin is tossed three times.

   (a)

   Find the probability that all three tosses are heads.

   [1 mark]

   • A$0.18$
   • B$0.216$
   • C$0.064$
   • D$0.6$

   (b)

   Find the probability of exactly two heads.

   [1 mark]

   • A$0.144$
   • B$0.288$
   • C$0.36$
   • D$0.432$

   (c)

   Given that at least one toss is a head, find the probability that all three tosses are heads.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   Events $A$ and $B$ are independent. $P(A)=2p$ and $P(B)=p$, and $P(A\cup B)=0.52$.

   (a)

   Show that $2p^2-3p+0.52=0$.

   [3 marks]

   (b)

   Solve the equation, justify your choice of root, and find $P(A'\cap B')$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A learner passes a driving test at the first attempt with probability 0.6. A learner who fails may take the test once more, and passes at the second attempt with probability 0.75. The results of different learners are independent.

   (a)

   (i) Find the probability that a learner passes within two attempts. [2]
   (ii) Find the probability that two learners both pass within two attempts. [2]
   (iii) Find the probability that, of two learners, exactly one passes at the first attempt. [2]

   [6 marks]

   (b)

   (i) Given that a learner passes within two attempts, find the probability that the learner passed at the first attempt. [2]
   (ii) Three learners take the test. Find the probability that exactly two of them pass at the first attempt. [2]
   (iii) Find the probability that at least one of the three learners does not pass within two attempts. [2]

   [6 marks]

   Total for question 8: 12 marks

End of questions