Find the probability of both selected balls are the same colour (replacement and adding) Question:
A box initially contains 1 RED ball and 2 WHITE balls all the same size. You select a ball at random and then return it to the box together with another ball of the same colour. The box now contains four balls. Again you select a ball at random from the box. Find the probability that both selected balls are the same colour.
What I think:
red/red + white/white = (1/3*2/4)+(2/3*3/4)
 A: That is almost okay.   Your presentation is lacking, but it is apparent you grasp the method and those are the right numbers to use.
This is the Law of Total Probability at work.

Letting $B_1,B_2$ stand for the colour of the first and second ball drawn; with $r,w$ the colour constants, then the sought probability is:
$$\begin{align}\mathsf P(B_1{=}B_2) ~=~& \mathsf P(B_1{=}r)\mathsf P(B_2{=}B_1\mid B_1{=}r)+\mathsf P(B_1{=}w)\mathsf P(B_2{=}B_1\mid B_1{=}w)\\[1ex]=~& \tfrac 13\tfrac 24+\tfrac 23\tfrac 34\\[1ex]=~& \tfrac 8{12} \\[1ex] =~& \tfrac 23\end{align}$$

Work on your presentation; it is important.
A: *

*The probability of taking 1 red ball in the 1st turn is $\frac{1}{3}$


Now we add another red ball in the sample,


*

*The probability of taking 1 red ball in the 2nd turn is $\frac{2}{4}=\frac{1}{2}$


Hence,the P(taking red out red ball twice)=$$\frac{1}{3}\frac{1}{2}=\frac{1}{6}$$
lly,P(taking red out white ball twice)=$$\frac{2}{3}\frac{3}{4}=\frac{1}{2}$$
P(taking  out red ball twice)+P(taking out white ball twice)=$$\frac{2}{3}$$
