| A simple result from probability theory: P(success after n attempts) = 1 − (1 − p)^n Where: p = probability of success in a single attempt
n = number of attempts The equation computes the probability of achieving *at least one success* across n independent trials. The term (1 − p)^n represents the probability that *all attempts fail*. So the equation simply states: Probability of success = 1 − probability of failing every attempt. --- ## What Happens as Attempts Increase Consider the limit: lim (n → ∞) [1 − (1 − p)^n] = 1, for any 0 < p ≤ 1. Since (1 − p) is less than 1, repeatedly multiplying it drives the value toward zero. So the probability of failure across all trials goes to zero. Which means the probability of *at least one success approaches 1*. In other words: If success has any non-zero probability, repeated independent attempts will eventually succeed with probability arbitrarily close to certainty. --- ## A Simple Numerical Example Let p = 0.001 (0.1% chance of success per attempt). Probability of success after n attempts: n = 100 → ~9.5%
n = 1,000 → ~63%
n = 5,000 → ~99.3% Nothing about the attempt improved. Only the number of trials increased. --- ## Two Levers The equation highlights two ways to increase the probability of success: 1. Increase p (improve the success probability per attempt)
2. Increase n (increase the number of attempts) Most people focus on the first variable. But many real systems are dominated by the second. Examples include: * scientific experimentation
* startup iteration
* evolutionary processes
* randomized algorithms In these systems, progress emerges from *large numbers of trials exploring a search space*. --- ## The Only Requirement There is one constraint: p > 0 Success must be possible. If the probability of success is zero, repeated attempts will never produce success. So the key strategic question becomes: Is the system structured such that success is possible at all? If yes, increasing the number of attempts dramatically improves the odds of eventual success. --- ## Why This Matters Many real-world processes are better modeled as *probabilistic search* rather than deterministic planning. When viewed this way, iteration speed becomes critical. Reducing the cost and latency of each attempt increases the number of trials that can be run. Over time, the probability of finding a successful outcome rises toward certainty. Mathematically: 1 − (1 − p)^n → 1 as n → ∞. That simple equation captures a surprisingly large fraction of how progress actually happens. |