Convergence of random variables

Ever wonder why flipping a coin repeatedly doesn't always predict the next flip?

Imagine you're flipping a coin and want to know if you'll eventually get a run of heads or tails.

Flipping a coin is random, but if you flip it enough times, the chances of getting a long run of heads or tails decrease. This is because the outcomes are independent and equally likely.

Example

You flip a coin 100 times. The probability of getting 10 heads in a row is (1/2)^10, but as you flip more, the chance of seeing 10 heads in a row again drops dramatically.

Remember this

The Borel-Cantelli lemma shows that with infinite flips, the probability of seeing a long run of heads or tails becomes very small.

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

Related concepts

the optional stopping theorem says about martingales and stopping times

Ever wondered why betting on a roulette wheel doesn't change your odds over time?

Markov's inequality

Ever wondered how math can predict unlikely events?

Dirac delta function

How can we predict random events with certainty?

KL divergence is always ≥ 0 and equals 0 only when P = Q exactly

Why can't we just compare two things directly?

Hidden Markov model

Ever wondered how you can predict outcomes without seeing all the details?

Lyapunov exponents measure: rate of divergence of nearby trajectories in a dynamical system

How do tiny differences grow over time in chaotic systems?

Swipe through more Machine Learning concepts

Open Pocket Polymath