Trying to understand better the maximum likelihood method, i've end up finding a very interesting marbles from a jar problem that gives intuition about why the MLE is used. BUT there's a step in the problem - probably referring to baye's theorem - that i cannot understand. Here it goes. Consider a jar with black and white marbles in a 3:1 proportion, but the predominant color is unknown. Aiming to find out $\theta$ - which is the probability of drawing a black marble - 3 marbles are drawn from the jar, with replacement. Let X be the number of black marbles in the sample. As X $\frown$ Binomial (3, θ) with $\theta = 1/4$ or $\theta = 3/4$ comes: 
And so,
Now, my question is where those probabilities came from ? (27/64,9/64,1/64...)
What would be the arrangement of the conditional probability fraction? like what would be the a's and b's? $\frac{P(a \cap b)}{P(b)}$
Any info on how to connect that to the MLE problem and estimators, would be very appreciated. Thanks!
Hint: Notice that given the value of $θ$ you can calculate the probabilities that you are asking with the binomial distribution. For example the random variable $X|θ=\frac14$ is binomially distributed with parameters $n=3$ and $p=1/4$. In symbols $$X|θ=1/4 \sim \mathrm{Bin} (n=3, p=\frac14) $$Thus, f.e. $$P\left(X=0|θ=1/4\right)=\dbinom{3}{0}\left(\frac14\right)^0\left(1-\frac14\right)^{3-0}=\left(\frac34\right)^3=\frac{27}{64}$$ and similarly the rest.