Flipping 5 tails in a row, it’s still a 50/50 chance to get a tail in the next flip?

Question

Flipping a tail is a 1/2 chance, but flipping 6 tails in a row is a 1/64, so if after flipping 5 tails, why is it incorrect to say that your chance of flipping another tail is now lower, like you’re “bound” to get a head? I know this is the gambler’s fallacy, but why is it a fallacy? I get that each coin flip is independent, but it feels right (as fallacies often do) that in consecutive flips the previous events matter? Please, help me see it in a different way.

In: Mathematics

Answer 1

The gamblers fallacy isn’t a true “fallacy” at all. It’s a legitimate heuristic that sometimes gets applied incorrectly.

In any case, the heuristic you’re using is correctly applied to something like a shuffled deck of cards where a card is drawn without random replacement and one asks “what is the probability of drawing the ace of spades?” Each time you draw a new card the deck *physically changes*. It goes from 52 cards to 51, to 50, and so on. The odds of drawing the ace of spades goes up with each trial because the trial changes the state of the deck. The heuristic is also correctly applied widely to things like earthquakes (the probability of an earthquake tomorrow goes up each day as the tension between plates increases), Russian roulette, and searching for lost items (as you search each room in the houses the number of possible places the missing item could be goes down and, therefore, your odds of finding it in the next room goes up).

The heuristic is misapplied with things like flipping coins) because **the act of flipping a coin does not change the physical characteristics of the coin**. Other misapplications include playing slot machines, buying lottery tickets, drawing from a deck of cards *with* replacement, etc.

So when you are wondering when and where to apply this heuristic, ask yourself if each trial or event changes the state of the system. And if it does change the system ask yourself *how* it changes the system. There is no rule here to follow other than to build a mental model of the system you’re talking about.