A series of experiments is carried out and in each experiment the only possible outcomes are 'success' and 'failure'. The total number of successes is denoted by X. State two conditions which must be satisfied for the distribution of X to be modelled by a binomial distribution. Gromit invites 11 friends to a party. For each friend, the probability that he or she will accept the invitation may be taken to be 3. Use a binomial distribution to calculate the probability that

(a) exactly nine,

(b) fewer than nine, of the friends will accept the invitation. Give a reason why a binomial distribution might not be a good model in this situation. (C)