Question using Central limit theorem In probability

Question

Can anyone give me some direction please? I was thinking of using the Central limit theorem, but I'm not quite sure how to do it.

The box has 3 balls in different colors:

1. yellow
2. red
3. blue

For times take a random ball out of the box and return it. (n is a large number).

Calculate or find the Upper Bound:

A. There is a color we have not taken out any of the times.

B. There is a color that we removed at least (4/9)* times.

Answer 1

For Part A: $3(2/3)^n - 3(1/3)^n.$ Probabilities each of Y, R, B ignored (not disjoint) less probabilities 2 of 3 ignored. (Inclusion exclusion.) So, for $n = 5,$ the numerical answer (using R as a calculator) is $0.3827$ to four places.

3*(2/3)^n - 3*(1/3)^n
[1] 0.382716

Simulation of a million 5-draw experiments illustrates, giving probability $0.3830 \pm 0.0010.$

set.seed(2020)
n=5
nr.u = replicate(10^6, 
                 length(unique(sample(1:3,n,rep=T))))
mean(nr.u < 3)
[1] 0.383044        # aprx P(Not all 3) = 0.3827
2*sd(nr.u < 3)/1000
[1] 0.0009722583    # 95% margin of simulation err