Suppose we have a Bernoulli-like process $P$. At each step a coin is tossed and the outcome ("success", "failure") is recorded. What differentiate $P$ from the standard Bernoulli process, is that we pick a probability of "success" uniformly at random in range $(1/2, 1)$ at each step before we toss the coin.
I'm interested in finding an upper bound on the expected number of trials until the first "success" is tossed.
What I have thought, if the probability of "success" is at least $1/2$, then at each step $P$ is more probable to stop than a standard Bernoulli process, therefore an expectation of a standard geometrically distributed variable bounds from above the expectation of steps until the first "success".
How can I make this claim formal?