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
Assuming it’s a fair coin, it doesn’t know what happened in the earlier flips. Once you pick up the coin to flip again, you’ve removed the previous state, so the last flip can’t possibly influence the next flip. You seem to know this consciously, but it feels wrong to you because of the five in a row.
The thing about randomness is that it’s random, but your brain isn’t always good at knowing whether data looks random. If a coin strictly alternated heads and tails, that wouldn’t be random. So you have to learn to expect a certain amount of clustering. Two or three in a row should happen very frequently, and five or six in a row should happen occasionally. Even some short alternation sequences should happen from time to time. That’s part of the randomness, and your brain will see little patterns that aren’t there, and will expect them to continue.
The other thing specifically about getting multiple consecutive flips the same is that you begin to suspect the coin isn’t perfectly random. It does become more likely at some point that the coin is biased or even tails on both sides, and in those cases it would be rational to expect the pattern to continue. But that’s not randomness anymore.
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.
I think you grasp the concept that each flip is independent well enough, but a thing to remember is that every distinctive series has the same chance of happening as any other, so HHHHHH is as (un)likely to happen as HTHTHT.
The chances of eventually getting 3 H and 3 T are higher because there are more routes to get that total, but only one route to get 6 H.
It’s just that each flip eliminates all the other routes that had a chance at the beginning.
I would like you to explain to me why you think this feels right? You clearly understand why the 50/50 chance is accurate. You explained it in your own post. To me it feels very wrong that the odds would be anything but 50/50, it would change my entire view of how the world works if that were not the case.
If you flipped a coin thousands of time and looked at every sequence that had 5 tails in a row than the sixth flip in that sequence would be fairly evenly divided between heads and tails.
The coin doesn’t know that the last flip was heads. Each event is its own event, with no link to previous or future events.
Each flip is its own completely unique event, and no other flip has anything to do with it, so it’s always 50/50.
Let us pretend that a machine flipped a coin 6 times and it recorded each result.
HTHTHT would be heads-tails-head-tails-heads-tails
HHTTHT would be heads-heads-tails-tails-heads-tails
Now this machine does this 6 flip and record exercise one million times, and you have access to that table.
If you go look for how many of the entries are TTTTTT (six tails) you will find that it is 1/64 of the total number of recorded entries.
Next you go look for every entry that starts with five tails, to TTTTT* where the * is either H or T. If you put all of those entries together you will find that all of them are 2/64 of the total of all recorded entries. And if you look at them the number of TTTTTT entries and TTTTTH entries are the same.
From an emotional point of view, it does feel like the heads is “due” to come up. But if it helps think of it this way “what are the chances of me flipping 6 tails in a row?” is a different question than “what is the chance of me flipping a tail, given that there were 5 flips before it?”
Each flip of a single coin is an independent event with a 50:50 chance for each option.
So, while the odds of flipping a coin, and having it come up “heads” 6X in a row is 1/64, each INDIVIDUAL flip is still a 1/2.
The odds of flipping 4 tails and then a head is 1/64. The odds of flipping 4 tails and then another tail is 1/64. So, once you’ve flipped the 4 heads, odds are even either way.
But, that assumes a “fair coin.”. After, say, 100 tails in a row, you might start thinking “this isn’t actually a fair coin.”.
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