Which coin looks more fair?

Coin #1 seems more fair because it gave a mix of heads and tails, instead of all tails… right?

But wait! For a fair coin, the chance of flipping Heads Tails Tails Heads is the same as the chance of flipping Tails Tails Tails Tails

Try flipping a fair (50/50) coin a few times with the buttons below.

But wait wait wait wait… a coin that always lands Tails is still unfair, right?

Probably. Wikipedia has the derivation, but if you only consider a specific type of coin (one flip doesn't influence another, fixed probability of getting heads), then if you flip one 4 times and get Tails Tails Tails Tails, you can be ~93% sure that the probability of getting Tails is not between 45% and 55% (in other words, not fair).

So which is it, fair or unfair?

When we look at the likelihood of sequences of four coin flips together, instead of treating it like a two-sided coin, we're actually treating it more like a 16-sided die. One face reads Heads Heads Heads Heads, another side reads Heads Heads Heads Tails, etc.

So while Tails Tails Tails Tails is somewhat rare for a single coin flipped four times, you have 1/16 chance rolling it with 16-sided die.

If you want to figure out if the coin is unfair by rolling the 16-sided die, you have to roll it many more times. Tails Tails Tails Tails Tails Tails Tails Tails Tails Tails Tails Tails Tails Tails Tails Tails Tails Tails Tails Tails…

P.S. Now you can dispel a common myth: getting a bunch of Tails in a row does not mean you'll get Heads next

If you flip a coin and get Tails Tails Tails, and someone says that you're probably going to get Heads next (because look at all those Tails!), you know that it's actually still 50/50. They're equally likely:

You can double-check by switching the graph above to "Three Tails First".