Find a repeat-draw distribution and approximate its sample mean

A bag has three identically shaped balls numbered 1,2,3. Draw one at a time at random with replacement. Let $X$ be the number of draws until any ball is drawn a second time; matching draws need not be consecutive.

(i) Show $P(X=4)=2/9$ and find the distribution of $X$.3 marks

(ii) Show $E(X)=26/9$ and find exact $\operatorname{Var}(X)$.3 marks

(iii) Using a suitable approximation, find the probability that the mean of 44 independent observations exceeds 3.3 marks