N2012 P2 Q9 [Modified]

In an opinion poll before an election, a sample of \(30\) voters is obtained.

1. The number of voters in the sample who support the Alliance Party is denoted by \(A\).
   State, in context, what must be assumed for \(A\) to be well modelled by a binomial distribution.[2]

Assume now that \(A\) has the distribution \(\mathrm{B}\left( 30,p \right)\).

2. Given that \(p=0.15\), find \(\mathrm{P}\left( A=3\,\,\mathrm{or}\,\,4 \right)\).[2]
3. For an unknown value of \(p\), it is given that \(\mathrm{P}\left( A=15 \right)=0.06864\) correct to \(5\) decimal places. Show that \(p\) satisfies an equation of the form \(p\left( 1-p \right)=k\), where \(k\) is a constant to be determined. Hence find the value of \(p\) to a suitable degree of accuracy. Given that \(p<0.5\).[3]