> two-thirds gambled on tails at some stage in the experiment
I'm not sure why that's called out. If you've just had 6 heads in a row the next 4 "should" be tails, so it's not irrational to bet on tails is it?
I'm not sure why that's called out. If you've just had 6 heads in a row the next 4 "should" be tails, so it's not irrational to bet on tails is it?
I realize you're probably joking, but since this argument is intuitively appealing to many people, I will answer as if it was serious: if you have a weighted coin that is 60% likely to land on heads, that means it's 60% likely to land on heads on any given toss. On the first toss. On the second toss. Any given toss. Even after you have tossed it 6 times and seen 6 heads in a row, the coin is still 60% likely to land on heads. The coin has no "memory". Previous results have no effect on future results.
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.
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.
So the situation described in that paper is that you are given the true odds of the coin, 60% heads. In this case it's just as I described - knowing previous results doesn't tell you anything useful.
> Whereas I'm only flipping 10 times, so I won't necesssarily get 60% heads.
This is true. In fact there is only about 25% chance of getting exactly 6 of the 10 to be heads (but nearly 70% chance of >= 6 heads). You can work this out with something called the binomial distribution. Chance of getting 10 heads in a row is .6%
A more interesting aspect is when you don't know the odds (or don't trust what you've been told). In this case it's definitely important what the history is. So given your 10 flips, we can ask questions like "how likely is it that this coin is fair (50/50) given the 10 flips I just saw".
It turns out the best estimation of the true probability is, pretty intuitively, (h+t)/h; this will jump aroudn for small N . In practice you are more often looking at something like P(0.55 < p < 0.65 | samples) , i.e. the probability that the true value lies between 0.55 and 0.65 heads, given the 10 flips I've seen).
Obviously in these cases, the more samples you have seen the tighter the estimate get. You can also ask questions like how many flips do I need to see to be confident at a certain the coin is really 0.6 heads.
You can still have streaks of hundreds, thousands, millions of either heads or tails in a row.
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.
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.
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.
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).
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
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?
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 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.
However, until such probability is established, if I see heads in a row - my intuition would tell me that the physics is skewed towards heads. I don't think that it would be unreasonable to think that in such circumstances until one gets a larger sample of throws.
> 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.
People rolling dice aren’t, even when they try to be, perfect randomizers, and with a maximally favorable result and an action which demonstrably repeats it, there’s a strong incentive to repeat the action as accurately as possible rather than even trying to be a perfect randomizer.
OTOH, the probability of some other explanation besides a fair coin isn’t consistent among all other possible sequences, so what the actual result does to your estimate of the likelihood of a fair coin depends on the actual sequence, and your basis for believing the coin was fair going in.
Things are only slightly different with, say, a coin you’ve been told has a 60% bias.
EDIT: For instance, if there is a 1:1,000,000 chance that you would be given an underestimate of bias and a 1:1,000,000,000 chance of the outcome you actually receive being true if the coin had only the bias you were informed of, its a lot more likely that you were lied to than that you just got an unusually consistent set of results.
It doesn't change the point of my original comment, regardless of the improbability of 60 heads in a row, you aren't "due" 40 tails in a row because the events are independent. That's all I was getting at before you took us on a weird tangent.
The original point of your comment is correct, at least from a probability standpoint. You don't get "owed" tails. I guess my hint was that there are sometimes other factors at play that mean the theory goes out the window. Like if someone shuffles a deck in front of you and it ends up new deck order, it's more likely they're a magician than lucky.
I remember one time when I rolled really low numbers on a D20, and then there was this really important roll, where I had to get a 20. I confidently said "No problem, I rolled a few really low numbers in a row, so this is definitely going to be a 20, it's pure statistics". Also throwing some calculation in there: "I rolled a 2 and a 1, so in 3 rolls I should get a total of 30 on average, so that means I actually still need 27 to reach the average. That results in more than 100% chance of rolling a 20 right now". And then I actually rolled a 20, was able to keep my cool and a straight face "see, it's just theory". Pure gold! LOL :D
Your other friend has been playing longer, before you even started. They saw 13 tails and then your 6 heads. The next throw should be heads to even it out for them.
Why is your history more of an influence than theirs?
The result of each flip is completely independent of what came before it. In your example the 7th flip is just as likely to be heads as the first flip, or any of the other 5 flips that landed on heads.
But most people would agree with the irrational bet. This tendency is known as the Gambler’s fallacy (https://en.wikipedia.org/wiki/Gambler's_fallacy).
That's not how this works. Each toss is independent, so you should never pay attention to previous results if you know the true odds.