Distinguish CLT estimates from exact probabilities and strict thresholds

(a) A random sample of size 50 is taken from $X\sim B(10,0.4)$. Find the probability that the sample mean is less than 4.5.

(b) $Y$ is normally distributed with mean 4 and variance 3, and $\bar Y$ is the mean of $n$ independent observations. (i) Find $P(\bar Y<5)$ when $n=5$. (ii) If $Y$ is not normal, state one condition for applying the central limit theorem to calculate this probability. (iii) If instead $Y$ is uniform on $(0,12)$, with $E(Y)=6$ and $\operatorname{Var}(Y)=12$, find the smallest sample size such that $P(\bar Y>5)>0.99$.