Markov's inequality

Ever wondered how math can predict unlikely events?

Imagine you're playing a game where you roll a die. You're curious about the chances of rolling a number greater than 6, even though a die only has numbers 1 through 6.

Chebyshev's inequality helps us understand that even though it's impossible to roll a 7 on a standard die, there's still a tiny chance that something unusual happens, like rolling a 6 repeatedly.

Example

Let's say you've rolled a 6 ten times in a row. Chebyshev's inequality would show that this is highly unlikely, but not impossible.

Remember this

Chebyshev's inequality tells us that extremely unlikely events can still occur, even if they're not guaranteed.

Text adapted from Wikipedia, licensed under CC BY-SA 4.0.

Related concepts

Hoeffding's inequality

Hoeffding's inequality bounds tail probability for sums of bounded random variables

Chebyshev's inequality

Chebyshev's inequality limits the probability of deviation from the mean

Chebyshev's inequality says: P(|X-μ| ≥ kσ) ≤ 1/k²

Chebyshev's inequality states P(|X-μ| ≥ kσ) ≤ 1/k²

the Dirichlet distribution does: distribution over probability simplices

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Chernoff bound

Chernoff bounds provide exponentially tight concentration inequalities

Entropy H = -Σ p(x) log₂ p(x) measures average surprise in bits

How do we measure uncertainty in everyday decisions?

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