Chernoff's inequality

Chernoff's inequality is like a magic spell that helps us figure out how likely something is to happen. Let's say we have a big group of toys and they can be either red or blue. We don't know how many are red and how many are blue, but we want to know how likely it is that most of the toys are red.

Chernoff's inequality lets us use math to figure this out. We can think of each toy as having a red or blue hat on. We count how many toys have a red hat and we call that number "X". Then we imagine flipping a coin to decide whether to keep or throw away each toy (this is called "sampling"). When we're done, we count how many toys are left and we call that number "Y".

The magic spell (Chernoff's inequality) helps us figure out how likely it is that Y will be close to X (meaning we didn't randomly throw away too many red toys; we kept most of them). We start by telling the spell what probability we want to be really sure of (like 99% or 95%). Then we tell it how many toys we have (let's say 100) and what fraction of them we want to be red (let's say 60%).

The magic spell does some math and gives us an answer. It tells us how many toys we need to keep (the number "Y") so that we can be really sure that at least 60% of the toys are red. If we keep enough toys (as told by the spell), then we can be very confident that most of the toys are red.

That's what Chernoff's inequality does! It helps us figure out how likely it is that most of the toys are red (or whatever else we're interested in) using a magic spell and some math.