Probability Calculations That Trip Up Even Experienced Developers

Probability feels intuitive until you actually have to calculate something. I have watched senior engineers confidently state that the probability of two independent events both happening is the sum of their individual probabilities. It is the product. And that single mistake can cascade through an entire risk model. Where probability goes wrong in practice The birthday problem is the classic example. In a room of 23 people, there is a greater than 50% chance that two people share a birthday. Most people guess you need around 180 people for even odds. The intuition fails because humans are bad at combinatorics. This same failure mode shows up in software engineering constantly. Hash collisions. You have a hash function that produces 2^32 possible values. You are hashing 100,000 items. What is the probability of at least one collision? It is not 100,000 / 2^32 (about 0.002%). Using the birthday problem approximation: 1 - e^(-n^2 / 2m), where n is the number of items and m is the number