Let X be the number of successes in 25 independent trials with probability 1/10 of success on each trial. Let m be the most likely value of S.

a) Find m.

b) Find P(X = m) correct to 3 decimal places.

c) What is the value of the normal approximation to P(S = m)?

d) What is the value of the Poisson approximation to P(S = m)?

e) Repeat a) for n = 2500 trials instead of 25. Which would now give the better approximation to P(S = m), the normal or the Poisson approximation? Find P(S = m) approximately using the best approximation.

f) Repeat e) for 2500 trials and p = 1/1000 instead of p = 1/10.