2007 H1 Q9 [Modified]

A random variable \(X\) has a binomial distribution with \(n=6\) and probability of success \(p\).

1. Given that \(\mathrm{P}(X<4)=0.25\), find \(p\).[2]
2. Given that \(\mathrm{P}(X=3)=0.08192\), show that \(p\) satisfies the equation \(p(1-p)=k\), where \(k\) is a constant to be determined. Given further that \(p<0.5\), find \(p\).[4]

It is now given that \(p=\frac{1}{4}\).

3. Find \(\mathrm{P}(X=4)\), giving your answer as a fraction.[1]
4. The mean and standard deviation of \(X\) are denoted by \(\mu \) and \(\sigma \) respectively. Find \(\mathrm{P}(\mu -\sigma < X < \mu +\sigma )\), correct to \(2\) decimal places.[5]