Modelling with probabilityEdexcel A-Level Maths: Revision notes

A model is a simplified description of a real situation, built from assumptions, that lets you calculate probabilities. Common assumptions are that a coin or die is fair (all outcomes equally likely), that trials are independent, and that the probability stays constant. Example: modelling two die rolls with a fair die gives $P(\text{sum}=7)=\frac{6}{36}$. The answers are only as good as the assumptions, and all the working in a model question is done inside the model.

Exam questions ask you to criticise an assumption and say what a more realistic one would change. Be specific to the context. Fair die or coin: real ones can be biased, so some faces are more likely. Independence: a common cause links events, for example traffic delaying consecutive buses, or siblings who travel together. Constant probability: it can vary with time, weather, opposition or form. A good answer names the assumption, says why it is doubtful here, and states the likely effect, for example 'the probability of all buses on time would fall if $p$ rises above $0.1$'.

A binomial model $X\sim B(n,p)$ needs: a fixed number of trials $n$, two outcomes (success or failure), a constant probability $p$ and independent trials. $P(X=r)=\binom{n}{r}p^r(1-p)^{n-r}.$ Example: $X\sim B(10,0.7)$ gives $P(X=7)=0.267$ and $P(X\ge8)=0.383$. A discrete uniform model suits a fair die, giving each value $\frac1n$. A discrete model given as a table is valid if its probabilities sum to 1. For a continuous variable, probabilities are areas under a curve. Check the conditions before you choose a distribution.

To test a model against observations, compute the expected result and compare it with what was observed. A fair die rolled 60 times gives an expected $10$ sixes; 18 sixes is far more than expected, so the 'fair' assumption looks doubtful. If the model gives a probability for the observed result (or something more extreme) that is quite large, such as $P(X\ge9)=0.149$, the result is not unusual and the model is not contradicted. If it is very small, the model is questionable. Small samples give weak evidence, so say a larger sample would help.