A fixed number, \(n\), of cars is observed and the number of those cars that are red is denoted by \(R\).
State, in context, two assumptions needed for \(R\) to be well modelled by a binomial distribution.[2]
Assume now that \(R\) has the distribution \(\mathrm B(n,p)\).
Given that \(n=20\) and \(p=0.15\), find \(\mathrm P(4\leq R<8)\).[2]
Given that \(n=240\) and \(p=0.3\), find \(\mathrm P(R<60)\) using a suitable approximation, which should be clearly stated.[3]
Given that \(n=240\) and \(p=0.02\), find \(\mathrm P(R=3)\) using a suitable approximation, giving your answer correct to 4 decimal places and explaining why the approximation is appropriate in this case.[3]
Given that \(n=20\) and \(\mathrm P(R=0\text{ or}\hspace{0.5em}1)=0.2\), write down an equation for the value of \(p\), and find this value numerically.[2]