If you see 60 heads in a row in the real world you've got a trick coin. The odds of that are 1/10^17.

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.

If you see 60 heads in a row from a coin you’ve been informed is biased to produce heads on average 60% of the time, you'd need a pretty strong bases for trust in your information to not conclude that the most likely explanation is that the bias was underreported. Yes, its possible with the reported bias (or even if the bias was overreported), but that's not the most likely conclusion absent some pretty firm external evidence of the accuracy of the bias estimate you were provided with.

> 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.

I don't believe you.

I mean, that's fine, it's an anecdote. If you'd like, take a few dice and set up cameras and an automatic rolling mechanism and see if there are any improbable sequences like alternation between two or three number or a long run of a single number, or a long run without a particular number appearing. Over enough trials you are likely to encounter these kinds of events.

There will always be improbable sequences; with a fair coin, every possible sequence of length N is equally improbable, after all; if you flip a fair coin 64 times, the sequence is guaranteed to be a 1 in 2^64 event.

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.

If you had a camera pointing at a thousand coins that flipped once every second since the beginning of the universe, you still would probably not see 60 heads in a row.

If you had flipped one coin 4.35e17 times and never saw 60 heads in a row, on a biased coin, I'd be rather surprised. (took 13.8 billion years as the age of the universe). Do that 1000 more times and still don't see 60 heads in a row it would be even more surprising.

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.

I did some miscalculations. 2^60 is 1.15E18. So you couldn't do a thousand times per second. But it probably wouldn't happen at 1 per second.

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 used to drive a fellow RPG-er crazy with this. Whenever I would roll a few times low numbers, I would say "Alright, next time will be high, that is obvious, it's pure statistics!". At first he would object, but still even after he knew that I knew, that statement would still drive him mad.

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