Edexcel A level Maths (9MA0) · Statistics › Probability
Practise Modelling with probability. 1 exam-style questions plus unlimited generated ones on this subtopic, at up to four difficulty levels, with full mark schemes and a progress tracker. Free, no account needed.
Written for this site in the style of Edexcel exam questions. They are not taken from real past papers.
Question 1Easy3 marks
The probability that a darts player hits the bullseye with any one throw is modelled as 0.2, independently of other throws.
(a) Using this model, find the probability that the player hits the bullseye with both of the first two throws.[2]
(b) State an assumption of the model that you used in your calculation.[1]
Show the answer and mark scheme
(a)Answer: \(0.04\)
M1 for \(0.2 \times 0.2\)
A1 for 0.04
(b)Answer: The probability of hitting the bullseye is the same for every throw (and throws are independent).
B1 for the probability of hitting the bullseye is the same for every throw (and throws are independent)
Question 2Medium4 marks
A weather forecaster models the probability that it rains on any given day in April as 0.45, independently of every other day.
(a) Using this model, find the probability that it rains on at least one of the 5 days from Monday to Friday.[2]
(b) Criticise one assumption of the model, and describe the likely effect of a more realistic assumption on the probability that it rains on both Saturday and Sunday.[2]
Show the answer and mark scheme
(a)Answer: \(0.950 \text{ (3 s.f.)}\)
M1 for \(1 - 0.55^{5}\)
A1 for awrt 0.950
(b)Answer: Weather tends to persist, so rain on one day makes rain on the next day more likely: consecutive days are not independent. So the probability that it rains on both days is likely to be higher than the model predicts.
B1 for a valid criticism of independence or of a constant probability, in context
B1 for a sensible, consistent statement about the likely effect
Question 3Hard9 marks
A model for the number of attempts a learner needs to pass a driving test assumes that: • the probability of passing at the first attempt is 0.45 • the probability of passing at any later attempt, given that all previous attempts were failed, is 0.6 • the learner keeps trying until they pass.
(a) Find the probability that a learner passes within three attempts.[2]
(b) Show that the probability that a learner fails each of the first \(n\) attempts is \(0.55 \times 0.4^{n - 1}\).[2]
(c) Find the smallest value of \(n\) for which the probability that a learner passes within \(n\) attempts is greater than 0.99.[3]
(d) Give two criticisms of the assumptions of this model.[2]
(b)Answer: Fail the first (0.55), then fail each of the next \(n - 1\) attempts (0.4 each).
M1 for failing the first attempt with probability 0.55 and each later attempt with probability 0.4
A1* for \(0.55 \times 0.4^{n - 1}\)
Worked solution: The first attempt is failed with probability \(1 - 0.45 = 0.55\). Each of the next \(n - 1\) attempts is failed with probability \(1 - 0.6 = 0.4\), given the previous failures. So the probability is \(0.55 \times 0.4^{n - 1}\).
(c)Answer: 6
M1 for \(1 - 0.55 \times 0.4^{n - 1} \gt 0.99\), i.e. \(0.4^{n - 1} \lt \frac{0.01}{0.55}\)
M1 for using logarithms, reversing the inequality because \(\ln 0.4 \lt 0\): \(n - 1 \gt 4.37\)
(d)Answer: e.g. the chance of passing is likely to change with each attempt (more practice, or more nerves), not stay at 0.6; different learners have different chances; learners may give up rather than keep trying.
B1 for one sensible criticism in context
B1 for a second, different criticism in context
Worked solution: For example: a learner's chance of passing is likely to change from attempt to attempt (more lessons and experience may raise it), rather than being 0.6 every time; and different learners have very different chances, so one set of probabilities cannot fit everyone.