Hoeffding's inequality

Okay kiddo, so let me tell you about something called Hoeffding's inequality. Have you ever flipped a coin before? When you flip a coin, you have a 50% chance of getting heads and a 50% chance of getting tails, right? Well, Hoeffding's inequality is kind of like a way to give us a rule about how accurate our good guesses are when we flip lots and lots of coins.

Let's say we want to make a guess about how many times we'll get heads if we flip 100 coins. We might guess that we'll get heads about 50 times, right? But there's a chance we could be wrong- maybe we'll really get heads 55 times, or even only 45 times. Hoeffding's inequality tells us how likely we are to be wrong when we make a guess like this.

It works like this: imagine we've flipped our 100 coins and we've gotten heads 50 times. We can use Hoeffding's inequality to figure out that we're very likely to be within 10% of our original guess- so, between 45 and 55 heads. This is why Hoeffding's inequality is really useful for people who are trying to make predictions or estimate how much of something there is. It helps us know how accurate our guesses are likely to be, even when we're guessing about things we can't see, like how many heads we'll get when we flip a coin a bunch of times. Cool, right?