Modelling with probabilityEdexcel A-Level Maths: Subtopic test

1. 1
   A fair six-sided die is rolled twice. A student models the situation by assuming that the two rolls are independent and that each face is equally likely on each roll.

   (a)

   Using the model, find the probability that the total of the two rolls is 7.

   [1 mark]

   • A$\frac{1}{12}$
   • B$\frac{1}{36}$
   • C$\frac{1}{6}$
   • D$\frac{1}{7}$

   (b)

   Using the model, find the probability that at least one of the rolls is a six.

   [1 mark]

   • A$\frac{11}{36}$
   • B$\frac13$
   • C$\frac{25}{36}$
   • D$\frac{1}{36}$

   (c)

   The die is rolled 60 times and a six occurs 18 times. Comment on the assumption that the die is fair.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A bus company models each of its buses as being late with probability $0.1$, independently of every other bus. Three buses run on a route each morning.

   (a)

   Using the model, find the probability that exactly one of the three buses is late.

   [1 mark]

   • A$0.081$
   • B$0.027$
   • C$0.729$
   • D$0.243$

   (b)

   Which statement describes the most likely weakness of the model?

   [1 mark]

   • AA probability of $0.1$ cannot be used for a real bus
   • BHeavy traffic that delays one bus is likely to delay the next, so independence may not hold
   • CThree buses is too few to apply a probability model
   • DA bus can only be late or not late, so a probability model is invalid

   (c)

   At rush hour, the true probability that each bus is late is greater than $0.1$. State the effect on the probability that all three buses are on time, and justify your answer.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The number of goals $X$ scored by a football team in a match is modelled by $P(X=0)=0.2$, $P(X=1)=0.35$, $P(X=2)=0.25$, $P(X=3)=0.15$ and $P(X=4)=k$, with no other values possible.

   (a)

   Find the value of $k$ and the probability that the team scores at least 2 goals.

   [3 marks]

   (b)

   The team plays three matches. Assuming the matches are independent, find the probability that it scores at least 2 goals in exactly two of them. Give one reason why the model may be unrealistic.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A school's records show that $70\%$ of its students travel to school by bus. A teacher chooses 10 students at random and models the number $X$ who travel by bus as $X\sim B(10,0.7)$.

   (a)

   (i) Find $P(X=7)$. (ii) Find $P(X\ge8)$. (iii) State two assumptions of the model, and explain why one of them may not be reasonable if the 10 students were all taken from one form group.

   [6 marks]

   (b)

   The teacher now chooses 10 students from Year 12 only, and 9 of them travel by bus. Find $P(X\ge9)$ under the model, and comment on whether this suggests that Year 12 students are more likely than the school as a whole to travel by bus.

   [6 marks]

   Total for question 4: 12 marks

End of questions