[Jtsummers]

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.

[dragonwriter]

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.

[travisjungroth]

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.

[Jtsummers]

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.

[travisjungroth]

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.