Comparing independent (geometric) trials vs Bayesian learning across repeated attempts. Adjust parameters to see how the models diverge.
Someone once told me: "If the chance of success is 1%, you just need to try 100 times and you're guaranteed to succeed." It sounds right. It isn't.
If each attempt is independent — that is, the outcome of one try doesn't affect the next — the probability of success doesn't add up linearly. What actually accumulates is the probability of not failing every single time. The correct question is: after n tries, what are the chances you've succeeded at least once?
The answer is 1 − (1 − p)ⁿ. For p = 1%, you need 458 tries to reach a 99% chance of success — not 100. And you can never reach 100%, no matter how many attempts you make. Each additional try chips away at the remaining probability of failure, but never eliminates it entirely.
That's the independent model. Things get more interesting if the attempts aren't truly independent — if each failure teaches you something and raises your probability on the next try. That's the Bayesian model. Use the tool below to explore both.
Each trial is identical. P(fail all n) = (1−p)ⁿ. Requires log(1−τ)/log(1−p) trials to hit threshold τ. Assumes no learning — every attempt starts from zero.
Each failure raises p_t toward p_max at rate α. P(fail through n) = ∏(1−p_t). You need fewer trials because you're getting better — but only if α > 0 and learning is real.