Let \(S\) be the number of successes in 25 independent trials with probability \(\frac1{10}\) of success on each trial. Let \(m\) be the most likely value of S.

1. find \(m\)
2. find the probability that  \(\mathbf{P}(S=m)\) correct to 3 decimal places.
3. what is the normal approximation to \(\mathbf{P}(S=m)\)  ?
4. what is the Poisson approximation to \(\mathbf{P}(S=m)\) ?
5. repeat the first part of the question with the number of trial equal to 2500 rather than 25. Would the normal or Poisson approximation give a better approximation in this case ?
6. repeat the first part of the question with the number of trial equal to 2500 rather than 25 and the probability of success as \(\frac1{1000}\) rather that \(\frac1{10}\) . Would the normal or Poisson approximation give a better approximation in this case ?

[Pitman p122 # 7]