Modelling with probabilityAQA A-Level Maths: Revision notes

Section 1

What a probability model is

A probability model is a simplified description of a real situation, using assumptions so that probabilities can be calculated. Typical assumptions are that trials are independent, that the probability of success is constant, that outcomes are equally likely, and that the number of trials is fixed. Example: a school models each of 600 students as absent with probability $0.06$ , independently. This is a binomial-type model. State the assumptions in context, for example 'whether one student is absent does not affect whether another is absent'.

Key termsmodelassumption

Exam tip

When asked for an assumption, say it in the context of the question, not just 'independence'.

Section 2

Working within a model

Once the assumptions are stated, use the rules of probability. For independent trials multiply; for different arrangements add. For three buses, each late with probability $0.15$ : none late is $0.85^3=0.614$ , exactly one late is $3\times0.15\times0.85^2=0.325$ . For a footballer who scores each penalty with probability $0.8$ independently: exactly two of three is $3\times0.8^2\times0.2=0.384$ , all three is $0.512$ , and at least two is $0.896$ . Remember the factor of 3 for the number of arrangements.

Key termsarrangements

Common mistake

Forgetting the number of arrangements: $0.15\times0.85^2$ is only one order of one late bus.

Section 3

Critiquing assumptions

To critique a model, take an assumption and ask whether it is realistic.

Independence fails when events share a cause: buses delayed by the same road closure, students infected in the same outbreak.
Constant probability fails when the probability changes with time, place or the group.
Replacement matters for small populations: from 10 counters (4 red), $\frac4{10}\times\frac4{10}=0.16$ with replacement but $\frac4{10}\times\frac39=0.133$ without. Say what the effect is on the answer: positively linked events make extreme outcomes (for example all three late) more likely than the model says, so the model underestimates them.

Key termscritiquepositively associated

Exam tip

Name the assumption, say why it fails in context, then state the effect on the probability.

Section 4

More realistic assumptions

A refinement changes an assumption to fit reality better. For penalties, a refined model has the probability depend on the previous kick: $0.8$ after a score, $0.6$ after a miss. Exactly two scores are SSM, SMS and MSS: $0.8\times0.8\times0.2+0.8\times0.2\times0.6+0.2\times0.6\times0.8=0.32$ . With all three, $0.512$ , the probability of at least two is $0.832$ , which is lower than the $0.896$ from the simple model. The refined model is usually harder to calculate with, so the decision is a trade-off: a simple model may be good enough, especially for large populations where, for example, ignoring replacement has little effect (for 100 counters with 40 red, $\frac{40}{100}\times\frac{39}{99}=0.158$ , very close to $0.16$ ).

Key termsrefinementtrade-off

Common mistake

Saying only that a model is 'not accurate'. State which assumption fails and which way the answer moves.

That's the notes covered.

Carry on to the next subtopic.

Exam questions on Modelling with probability

A school with 600 students models each student as absent on a given day with probability $0.06$ , independently of every other student.
Suggest one change that would make the model more realistic, and give a reason.2 marks
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 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
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.
Find the probability that both counters are red (i) using the student's model, (ii) using the actual procedure.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).