Comment by blowski

If they said P(heads) is 60% and you get 4 tails in a row, you also might think the people running the experiment lied to you, especially if it happens near the beginning. But there’s a 13% chance in any sequence of four tosses.

but that means you should bet into the bias, not against it.

Yes, you are wrong, but your confusion is very common. It's so common it even has a name "The Gamblers Fallacy".

Over the long run, you expect 40% tails, but if you run the experiment an infinite amount of time there will be sequences of all-heads or all-tails.

Because the events are independent, the previous flips don't change anything about what happens next.

Let's say you're throwing a piece of paper into the trash can from a short distance. Suppose you can successfully throw the paper in the trash 100% of the time. You move your hand. Your hand moves the paper. Gravity pulls the paper down. It collides with the trashcan. It's just a bunch of physical objects exerting forces upon one another.

Now, suppose you keep throwing, but somebody has opened a window, so now there's an occasionally gust of wind, which moves the paper in unexpected ways while the paper is in the air. Now you no longer hit 100% of your throws. Sometimes the paper lands in the trashcan, sometimes you miss. Regardless, the paper is still only affected by physical forces: your hand, gravity, wind.

Now, suppose you've been really unlucky the past few throws: you have missed 5 throws in a row because of the darn wind. Does it make you more likely to win the next throw, because you are "due" a win? Of course not, because the wind doesn't know or care about your paper throwing hobby. The wind does what it does, regardless of how many of your throws landed in the trashcan. If anything, missing 5 throws in a row makes it _less_ likely to land the next shot, because it may indicate conditions unfavorable to throwing (strong wind, loss of confidence, etc.)

Now, the coin flipping experiment with the weighted coin obeys the same physical laws as the paper tossing experiment. It's just a physical object that's affected by forces from your hand, gravity, air, etc. If you throw 6 heads in a row, there's no magic that somehow alters the coin's path in the air on the 7th toss to make it come down tails. The universe doesn't care about our little games.

There are a few other nice answers here, but I think it's important to attack it from as many angles as possible.

The intuition that you're going for is that if the true rate is 60% heads and you've seen more than that then to hit 60% odds you _must_ have some extra tails _eventually_. Interestingly, that isn't actually required to make the odds work out to 60% eventually. I'll try for an intuitive explanation:

Say you've gotten 10 heads in a row but that the coin really only has a 60% chance of coming up heads.

- After 1000 extra flips you'll have 610 heads and 400 tails total on average for a 60.4% chance of heads so far.

- After 10k extra flips you'll have 6010 heads and 4000 tails for a 60.04% chance of heads so far.

- After 1M extra flips you'll have 600010 heads and 400k tails for a 60.0004% chance of heads so far.

Notice how the average percentage of heads is getting closer and closer to 60% even though the extra flips don't have _any_ bias toward tails. A temporary bias toward tails would _also_ suffice, and in much less time (some games like WoW use this for their loot tables I think), but it isn't necessary, and in the example of independent coin flips it does not happen.

> If I’m expecting 60% of my flips to be heads, and I’ve already had 60%, isn’t it more likely that the next one will be tails?

Nope.

> I’m sure you can probably tell I know next to nothing about either maths or probability, so feel free to explain why I’m wrong.

Lots of people have explained in terms of independence, which is correct. Another way of looking at it (definitely not more correct, but maybe more compatible with the “a series should eventually match the quoted probability” thinking) is in terms of infinity:

If you are expecting 60% of results to be heads, you expect that to hold over an infinite series of flips.

If you see any finite number of heads in a row, the probability for each of the remaining flips in the infinite series to get the total to 60% is…still 60%.

No finite series of results can change the probabilities necessary to get the infinite series to turn out as expected.

You’re talking about reversion to the mean, which is a phenomenon that’s related to the law of large numbers.

Law of Large Numbers says that, over an arbitrarily large random sampling size, you will eventually end up with a sample that perfectly fits the probability distribution.

But the probability of each individual sample is random. This means that, if each sample is randomly-selected and independent, your history of N samples does not affect your N+1th sample.

The regression to mean curve is only predictable in the big picture, each bump is 50/50 (or 60/40 in this case).

It's not that you're expecting 60% of your flips to be heads, but the coin has a 60% probability of being heads.

The former implies that previous flips have an effect on future flips. Or that, if you land on heads 6 times in a row, then the probability of it landing on tails goes up. How would a coin that's weighted to increase the odds of it landing on heads, somehow start landing on tails more frequently?

If you flip a normal coin and it lands on heads 10 times, you still have a 50% chance of getting heads the 11th time. The odds of it landing on heads 10 times in a row in the first place is vanishingly small (0.5^10 or 0.097%). But if it Does, the 11th flip still has a 50% chance. The first 10 flips don't affect the 11th. Physically, how Would the first 10 flips affect the 11th?

This is all assuming that the coin flips aren't somehow magically linked or casually dependent on each other. The math changes if the previous coin flip could somehow affect the next one. But in a situation where every single roll of the dice is purely independent, then by definition (Because they are Independent ) a previous roll doesn't have an impact on future rolls

You expect 60% of your flips to be heads at the outset. Let's say you flipped it a bunch and you're running at some rate.

How could the past flips of the coin possibly influence the flips you get in the future? The coin hasn't changed, the surrounding area hasn't changed, why would the coin suddenly have a different chance of turning up heads on your next flip? There's no probability god that mucks with random chance to make sure 'runs' are balanced overall. Every coin flip is independent, which means all the coin flips are also independent of the past coin flips.

If you've "had 60%", that means you've had an unlikely run of heads. Let's say the last 6 flips were 5 heads and a tail, a slightly unlikely outcome (3 in 16, I think). What physical force is acting on the coin to make it less likely to be heads, in the future? Why wouldn't it still have a 60% chance of coming up heads on the next flip?

It's always exactly 60%, no matter how many heads you have already had. That is pretty much by definition, since the problem states that the chances of heads are 60%.

In fact, in the real world getting an unlikely string of heads (or tails, or sixes, or whatever) outside of a casino setting probably means that the coin/dice/whatever are unfairly loaded and you should adjust your expectation for the next coin toss even further towards heads.

I upvoted you because, while you aren't correct in your assessment, I think this is a good "teachable moment". Human intuition about statistics is really, really bad.

I think people who have a better-than-average understanding of statistics forget how bad their intuition is. I suspect it leads to a lot of incorrect assumptions about what a "rational" behavior for someone working from only their statistical intuition would be.

The fact that your previous 6 flips were all heads was an unlikely outcome, but the coin has no recollection of what just happened and doesn't "care" about the past when you flip it again. The maths term for this is to say that each coin toss is "independent." I would not bet that you'd get another 6 heads in a row, but I would bet that the next coin flip will be heads.

Previous tosses do not change the outcome of subsequent tosses, so no, it’s not more likely to be tails, it has a 60% chance of being heads.

It's certainly low odds, but it's not impossible nor does it require a trick coin. I've seen people roll a 20 on a d20 10 times in a row, and then not a single 20 the rest of the session on the same die. Shit happens, it's probability and it may be improbable but it isn't impossible.

Thought experiment: in what way has it landed heads 60% of the time? It landed heads 100% of the trials so far, but the coin has no way of keeping track of that.

That's not a guarantee for any number of flips. For example, if you only flipped the coin one time, what does "60% of the time" even mean in that context? As your other replies have indicated, this is getting at the long-run frequency, meaning as you flip the coin more and more times, approaching infinity, the number of heads approaches 60%.

The key here is that it's expected to land heads 60% of the time. Take a normal coin, which is expected to land heads 50% of the time. If you flip a heads, do you instantly expect it to be tails next time? By your logic it would be impossible to ever flip heads twice in a row. Coins as a general rule aren't impacted by previous flips.