Modelling with probabilityAQA A-Level Maths: Subtopic test

1. 1
   A school with 600 students models each student as absent on a given day with probability $0.06$, independently of every other student.

   (a)

   Which of these is an assumption made by the model?

   [1 mark]

   • AStudents from the same family are more likely to be absent together
   • BThe probability of absence is higher in winter
   • CExactly 36 students are absent every day
   • DWhether one student is absent does not affect whether another is absent

   (b)

   During a flu outbreak that spreads between students, the probability of at least 60 absences on a day is likely to be

   [1 mark]

   • Agreater than the model suggests
   • Bless than the model suggests
   • Cexactly the same as the model suggests
   • Dimpossible to compare with the model

   (c)

   Suggest one change that would make the model more realistic, and give a reason.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A bus company assumes that each bus on a route is late with probability $0.15$, independently of the other buses. Three buses run on the route in the morning.

   (a)

   Find the probability that none of the three buses is late.

   [1 mark]

   • A$0.386$
   • B$0.614$
   • C$0.45$
   • D$0.85$

   (b)

   Find the probability that exactly one of the three buses is late.

   [1 mark]

   • A$0.45$
   • B$0.386$
   • C$0.325$
   • D$0.108$

   (c)

   A road closure delays every bus on the route. Explain why this makes the independence assumption unrealistic, and say how the true probability that all three buses are late compares with $0.15^3$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A bag contains 10 counters, 4 red and 6 blue. A student wants the probability that two counters taken at random from the bag are both red, and models the draw as if the first counter were replaced before the second is taken. In fact the first counter is not replaced.

   (a)

   Find the probability that both counters are red (i) using the student's model, (ii) using the actual procedure.

   [3 marks]

   (b)

   Another bag contains 100 counters, 40 of them red, and two are again taken without replacement. Find the probability that both are red. By comparing the difference from the student's model here with the difference in (a), comment on when the student's model is reasonable.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A footballer takes three penalties in a shoot-out. Model 1: each penalty is scored with probability $0.8$, independently of the others. Model 2: the first penalty is scored with probability $0.8$; each later penalty is scored with probability $0.8$ after a scored penalty and $0.6$ after a miss.

   (a)

   Using Model 1, find the probability that the footballer scores (i) all three penalties, (ii) exactly two penalties, (iii) at least two penalties.

   [6 marks]

   (b)

   Using Model 2, find the probability that the footballer scores at least two penalties. Compare your answer with part (a)(iii) and explain why Model 2 may be more realistic.

   [6 marks]

   Total for question 4: 12 marks

End of questions