is it reasonable to consider a binomial random variable as a distribution conditional on a Bernoulli random variable?

Question

Let $X$ denote a Bernoulli random variable represent the result of tossing one fair coin, one coin, one time.

Let $Y$ denote a binomial random variable represent the result of tossing one fair coin $2$ times.

Is it reasonable to consider $Y$ as a random variable conditional on the probability of head in $X$ is $0.5$?

Answer 1

No, that is not how a binomial random variable is interpreted, since the probability of success in each trial is typically considered a fixed, non-random quantity. When we talk about conditional random variables, we condition on something random.

You could, in theory, consider some model in which the probability of success of each trial is itself a random variable, but you would no longer have a binomial distribution in that case.