I learned that we can use $Beta$ distribution to assign a random variable representing the probabilistic distribution of probability. For simplicity, let's assume that we flip coins with unknown probability of success, denoted $B$.
Assuming that we don't know the initial value of $B$ we can set the prior probability to be $$B \sim Beta(\alpha, \beta)$$
Knowing B, the number of Heads in an experiment with $n$ trials, with known $p$ is going to be: $$X|B \sim Bin(n, p)$$
The question is how does the r.v. $1-B$ connect to the Binomial distribution in this case ? Using a change of variable, can be proved to be $f_{1-B}(x) = \frac{1}{Beta(\alpha, \beta)}x^{b-1}(1-x)^{a-1}$ and since $\frac{1}{Beta(\alpha, \beta)}$ is just the integration constant, $\alpha$ and $\beta$ could be interchanged (or use the connection to Gamma), this could lead to $Beta(\beta, \alpha)$
But is $1-B$ a random variable representing the probabilistic distribution of probability of a failure in an experiment ? Is there a way to connect it to the Binomial distribution ? Intuitively it makes sense, especially starting with the expected value of Beta.