There was previously a long extended debate between me and a couple other members of the forum, who haven’t posted in a while, where they maintained that a fair coin is more likely to flip tails if it has flipped heads three times in a row. I produced a simulation proving that isn’t the case, and they found creative ways to ignore that.

In any case, the question has come up on stack overflow with perhaps some enlightening answers, so I’m just posting this more as a reference in case it ever comes up again.

If you look at a poker game broadcast, they show the changing odds for any given flip of a card to be any given card, depending on what has already been flipped.

A deck of cards has a hard set distribution of results.

A coin flip works rather on a statistical basis. 4 tosses are less certain, with not terrible odds of getting 4 heads. 1000 tosses are likelier to give something closer to 500 heads and 500 tails.

If a coin has been tossed 1000000 times, and 700000 have been heads, the next flip is X% likelier to be tails than heads, such that that flip contributes to the expected 700000 heads out of 1400000 flips. Even though the 1000001 tosses aren’t hard distributed like a deck of cards, there is a soft expected distribution, thus every flip in a series or deck of flips changes the probabilities of the next. This is true in a soft, statistical sense rather than a hard, determined sense.

You can also count patterns within set series within the larger series and make mathematical predictions of the likely following smaller series.

This is because 50% means a soft expected 50 out of 100, scaleable to any number, with greater predictability the larger the number.

This is why if somebody rolls a long string of favourable results with dice in a dice game, loaded dice are suspected.

If I pick up a coin and flip 3 heads in a row, what do you think the probability is that the next one is heads? Assume it’s a fair coin, not double sided. Is it 50/50 or something different?

The problem is that the goal posts are shifted from “There is a 50-50 chance of a coin landing on heads or tails on a single flip.” to “After landing on heads 4 times in a row, are the chances greater that the next flip will land on tails?”

There is two different goal posts there:

On a single flip what are the odds.

On multiple flips, how many will land on heads and how many will land on tails.

As PZR explained, the more flips to the data the more it will be closer to a 50-50 average.

So for a million flips one should expect roughly 500,000 heads and 500,000 tails.
If on the last flip of 1 million flips the score is 700,000 heads and 299,999 tails, one would expect it to be a greater odds of being tails, because we know that on average 50% of the flips will be heads and 50% will be tails, and tails are long overdue, according to previously tested percentages, which have already been tested and found to be true.

The odds are 50-50 for a single flip. The odds of multiple flips are 50-50. So there is a 50-50 chance the coin will land on heads for that single flip. There is also the fact that out of 4 flips, 3 have already been heads, so that is already at least 75% of 4 flips, which we already agreed that out of 4 flips 2 should be tails to be 50-50. So either you are claiming the 50-50 of 4 flips is nonsense, or you must admit that a tails is more likely now that 3 of 4 have been heads. Which is it?

Except this is false. And if it were true, then one could win at casinos rather easily. Just track red and black on the roulette wheel. Eventually there will be an imbalance and you can start betting on the color that hasn’t been coming up. Even with the non-red black possibilities, you’d defeat the house. But you wouldn’t and I encourage doubters to try this.

Think about the odds of the falls at based on cause and effect in matter. They are not based on the past. The past is not causal in that way. You flip the coin that that object has a 50/50 chance to land on either. There is no wave or vibration or cause from previous tosses where is it? in what matter is it? does it matter if you wait two days for your next toss? How is the matter conveying this cause, this demand for balancing to the coin, tossed in the air?
Well, it’s not. We are dealing with an utterly new situation.

All around the world coins are being tossed, why should the coin you are tossing be affected by whatever imbalance there is in the head and tails totals. There is no cause and effect chain aimed for balance heading like a line of dominoes at your coin.

You just have this physical object with its shape and built in tendencies and those tendencies lead to 50/50 regardless of what went before. Whether you have a run of heads and immediately toss again. Or if you wait a month or ten years. I mention these different times to help us see that there is no cause somehow travelling through the either trying to get the coin to be close to exactly 50/50 over long stretches.

Try to find that cause in your model of the event. Where is it? How is it humming through matter?

We both agree that a single toss has a 50-50 chance of landing on heads or tails.

If you are sticking to that claim then you don’t have the right to talk about this entire scenario of adding the “3 previous flips” because to you every flip is 50-50, period.

So if I asked you out of 100 flips how many will be heads and how many will be tails, what is your answer?

If you have data on a million flips of a coin and you know that flips are equally divided between heads and tails in the data, then a series of heads in the data is more likely to be followed by a tails.

That may be what @Motor_Daddy is talking about with his two scenarios: one where you’re flipping a single coin once, and another when you have a bunch of data that you know some global fact about.

I think the mistake people make is assuming that the second situation holds just by virtue of it being a fair coin. That isn’t true.

I think you may be confusing me with someone else. I wasn’t in the discussion before that post, so it would be very strange if you asked me questions or had managed to predict I’d move the goal posts, which I didn’t do, in any case.

I see no effort to interact with what I wrote or to explain what the causal mechanism is that affects the coin in further throws.

What causes the next flip to be more likely to be tails? What is the physical force on the coin on that last toss that makes it more likely to be tails?
If you pause for an hour? Is it still more likely to be tails?
Is that next coin toss affected just but the tosses on that coin, or is it affected by tosses on all coins? If not the latter, why? What is the pressure to even out things on that coins and what is the name of the force?

If it supposedly happens in the next four tosses, that there is a greater chance now that more than two tosses of four will be tails - than there would be if the prior tosses had been equal - what is the force/cause of that shift? Is it stored as some kind of tendency in the matter of the coin?