# Is getting 100 Heads in a row from a fair coin a miracle or not?

Suppose a man continues to toss a coin until he gets 100 heads in a row. Suppose the outcomes of all tosses from the 9999901th toss to the 10 millionth toss are all heads and 100 heads in a row didn't happen before this.

Now, if someone went back in time or God told the man at the time he was going to start tossing the coin, that it would take him 10 million tosses to get 100 heads in a row, then it would not seem like a miracle to him because the probability of getting 100 heads in a row in a million tosses is already very close to 1. So, 100 consecutive heads was already bound to happen in 10 million tosses anyway.

But from the point of view of the man just before the 99999001th toss, he knows that the previous 9999000 tosses don't matter at all. The probability of getting 100 heads in a row at that moment is the same old (1/2)^100 (so it is almost an impossible event). The fact that he has been tossing the coin for a long time is not going affect his current chances of getting 100 consecutive heads. So, it is indeed a miracle that he got 100 heads in a row after that moment.

Whether an event is a miracle or a non-miracle, does it depend on the point of view?

• Miracle" is an event that is not explicable by natural causes alone." The laws of probability (assuming that they apply) do not rule out that event; thus, it is not a miracle. Seemingly, the best explanation is that the coin is loaded or there is some hidden "force" acting on it. Nov 22 '17 at 10:43
• If you toss a coin 10 million times, how did you calculate the probability of getting 100 heads in a row to be almost 1? Also, perhaps look into the Sleeping Beauty Paradox. Among other things, this is about whether and how probability/credence/certainty can depend on your viewpoint. It doesn’t concern the question of miracles, but I think that may be an advantage … Nov 22 '17 at 11:49
• One of the dictionary meanings of "miracle" is "an extremely outstanding or unusual event, thing, or accomplishment". In this sense 100 heads in a row, in everyday context, is a miracle. But then million tosses are not everyday. There is an old joke that after 10 heads in a row a mathematician cites the law of large numbers, but a physicist conjectures that the coin is biased. Our common sense is (rightly) closer to the physicist's. In everyday situations, where a biased coin is a possibility, it is more likely than a long random streak. Nov 22 '17 at 23:24
• As an aside, your numbers are off -- in 10 million coin tosses, on average you will only have less than 10^(-23) streaks of 100 heads. (and that's even counting a streak of 101 heads as two separate 100-long streaks, and so forth)
– user6559
Nov 23 '17 at 16:18
• Is this similar to the following puzzle? Imagine a lottery where only one person among all players win. There's a billion players. From the outside, it's no miracle that someone wins, but for the winner, this is miraculous that s/he won and noone else. Nov 23 '17 at 17:13

## 9 Answers

Yes, but only because every possible sequence of flips is a miracle.

Consider. Suppose I flip a fair coin 10 times in a row. Say I get HTTHTTHTHT.

Now that seems pretty normal, nothing remarkable about it.

But the odds of that exact sequence of flips are 1/1024. To see this, what are the odds the first flip is H? 1/2, right? What are the odds the second flip is T? 1/2. And the odds that the third flip is T? It's 1/2.

Continuing like this, we see that the odds of getting the sequence HTTHTTHTHT are 1/1024.

The odds of getting any particular sequence of ten flips are 1/1024. If you happen to get HHHHHHHHHH you'd say, "Wow, an amazing coincidence!" But if you got HTTHTTHTHT you'd say, "Well that was boring, nothing special about that." But it is special. It's just as special as all heads.

Do the same analysis for 1 million coin flips and you see that any particular sequence has exactly the same probability, namely 1/2 to the millionth power. Getting all heads or all tails or some completely random-looking string have exactly the same probability.

The reason random-looking strings happen more often is because there are so many of them! But any particular one has the same tiny probability and is a miracle. In other words you are comparing the one outcome HHHHHHHHHH to the entire huge collection of all the outcomes involving mixed heads and tails. Of course the mixed strings are more likely, but only because there are so many of them.

But flipping coins is nothing. The universe explodes into existence. Matter coalesces into the Milky Way galaxy, the sun and the planets. Life evolves on earth. Humans appear. Several hundred thousand years go by. A sperm fertilizes an egg. And here you are. Now that's a miracle.

You personally are a statistical miracle. Your existence is unimaginably unlikely yet here you are. One needn't adopt a theological point of view to see that everything in the world is truly a miracle.

• That opening statement is one of the best one-line explanations of a complex topic I've ever seen. Every case is a statistical impossibility, it's just human nature that attaches some sort of special meaning to 100 of the same thing, even though it's just as likely as any other arrangement Aug 19 '20 at 21:05
• +1 Excellent answer. The first line synthesizes it all. Nov 24 '20 at 18:05
• I think that it has more to do with the combinatorics of the coin flip, permutations describe probability in terms of order.
– yolo
Nov 29 '20 at 18:19
• It's important that the sequence HHH is of prior interest to humans. All of humanity combined could not produce a sequence of H and T that will resemble that one produced in secret by 100 flips of a coin. To the universe, there are no miracles, but to people, maybe. Nov 29 '20 at 19:54

If you're talking about the final toss in a sequence of 100, where 99 have already come up as heads, then the odds are STILL 50%. Whether the tosser (sorry, we may have to come up with a better name for him) realises this or not is largely due to the fact that his mind is geared towards pattern matching.

The human mind is very good at finding patterns in nature and then trying to ascribe some meaning to them. In short, we don't tend to believe that patterns can be random, but they can be every bit as probable as sequences that don't have an obvious pattern to them.

Someone trained in probability mechanics will see that. Someone without that training will struggle with the need to find meaning behind the pattern, even though (in this case) there isn't one. This is a challenging point for many and (arguably) is how faith is born; not just in any specific religion, but more in the idea that there is a power behind the patterns that we don't understand, and therefore can't understand.

In short, it may look like a miracle, but so do many things we see in nature until we apply a scientific method to them.

• I think the point where the OP is talking about tosser reflecting on probabilities is after he flips a tail just before starting his streak of 100 heads, not after the 99th head.
– user6559
Nov 23 '17 at 16:09
• True, but the point is, that with a proper understanding of maths and probability distribution in particular, the tosser would know that this outcome is equally as probable as any other combination in 100 tosses. The reason why this one stands out is because it's more recognisable as a pattern than many of the other combinations. This is because our minds are very good at seeing patterns first and seeking the cause next. In this case, the correlation of outcomes in sequence has nothing to do with causality. Someone who doesn't know this may see it as a miracle, those with such training won't. Nov 24 '17 at 1:37

David Corner describes two kinds of miracles.

1. Violation miracles are violations of natural law. However, getting 100 heads in a row does not violate any law of nature. It is merely unexpected. Hence it would not be a miracle.

2. Coincidence miracles are not violations of natural law but they have to be "religiously significant". In this example God predicted that he would get 100 heads in a row. This may be enough religious significance for the man to consider it a miracle. For someone else it may not.

This is not an exhaustive set of definitions of miracle and Corner claims:

In sketching out a brief philosophical discussion of miracles, it would be desirable to begin with a definition of "miracle;" unfortunately, part of the controversy in regard to miracles is over just what is involved in a proper conception of the miraculous.

The OP asks:

Whether an event is a miracle or a non-miracle, does it depend on the point of view?

If it is a coincidence miracle then the point of view of the one experiencing the miracle needs to be taken into account. That person needs to see the event, which is not a violation of a natural law, as religiously significant.

Corner, D. Miracles. Retrieved on April 26, 2019 from Internet Encyclopedia of Philosophy at https://www.iep.utm.edu/miracles/

Whether an event is a miracle or a non-miracle, does it depend on the point of view?

I say yes.

The heads-or-tails pattern is never a miracle in the sense that the result requires divine intervention as its explanation. Each result is adequately explained by the coin and its surroundings. The specific pattern HTHHT is no more or less likely than the pattern HHHHH.

But 100 H's sure does look like a miracle, given its almost otherworldly rarity. So, yes, the perception does indeed depend on where the tosser is in the process.

Obviously, it could meet some senses of the term, dependent on your waging.

a remarkable event or development that brings very welcome consequences. "it was a miracle that more people hadn't been killed"

But

Hume defines a miracle as 'a violation of the laws of nature', or more fully, 'a transgression of a law of nature by a particular volition of the Deity'

Not sure of any philosopher who defines it to include unlikely events, etc..

Either way, very unlikely events (10,000,000,000,000,000... heads in a row) are not Hume's miracles, because they are physically possible, just very unlikely.

I quite like the question, because it may raise the question why anyone at all should believe in miracles, given that we don't know any exceptionless regularities. I suppose the idea is that it is more likely that "100 heads in a row" is a miracle, than it just happened.

Not sure how that links to theism etc..

HTH though.

Whether an event is a miracle or a non-miracle, does it depend on the point of view?

No, of course not! -- tho when stated like that, your question kinda implies that we were looking from, yes, different points, but at the same event.

Which is not what your example described.

In your example, we are looking at two different events: getting a certain result out of 10,000,000 throws vs getting it out of 100 throws.

Is getting even one head in a row a muscle? If the coin has no reason to exist on it's own, but nevertheless does exist, and follows a set of unknown laws of nature such as gravity which has never been measured or even seen (only it's effects but not it itself) then how can it not be a miracle?

I think it would be a poor assumption to say that miracles are solely based on the rarity of an event. Walking on water is considered a miracle (assuming no trickery) due to the fact it's impossible, not because it's rare.

It would be subjective. I would say that minds are more interested in combinatorics than permutations (i.e we're more interested in how many heads in comparison to tails as opposed to the order in which they are gained). In this sense, the chances of getting full heads or tails is least likely in 100 throws.

If it provides a surprise to the flipper then by a subjective perspective it's a miracle.

This may provide another analysis:

https://www.jstor.org/stable/j.ctvzgb9fn.10?Search=yes&resultItemClick=true&searchText=%28miracle%29&searchUri=%2Faction%2FdoAdvancedSearch%3Fgroup%3Dnone%26q0%3Dmiracle%26q1%3D%26q2%3D%26q3%3D%26q4%3D%26q5%3D%26q6%3D%26sd%3D%26ed%3D%26pt%3D%26isbn%3D%26f0%3Dall%26c1%3DAND%26f1%3Dall%26c2%3DAND%26f2%3Dall%26c3%3DAND%26f3%3Dall%26c4%3DAND%26f4%3Dall%26c5%3DAND%26f5%3Dall%26c6%3DAND%26f6%3Dall%26acc%3Don%26la%3D&ab_segments=0%2Fbasic_SYC-5187_SYC-5188%2Fcontrol&refreqid=fastly-default%3Af513c86c723c3b8ec3f964858096a0d6&seq=1#metadata_info_tab_contents

It either means one of two things:

1. Something very unlikely happened
2. The experiment was flawed

Whatever you think a "miracle" is, it's a subset of (2). More likely the coin was weighted or the person flipping the coin somehow cheated. It might mean the universe was programmed to confuse us by a yet-unknown force, which you could consider a miracle.

People are more worried about how the universe and life exists at all - that's certainly unlikely on the conceptual order of 1/2^100 (if not much worse), and the explanations for how this happened tend to not be satisfying. You could define a miracle as something of interest which has an unobservable cause, in which case either we'll know why we all exist, or it was a miracle but we'll never be sure (because if we were sure, that would be too much knowledge for it to be a miracle).

The halting problem and Godel's theorem in mathematics do prove the existence of unknowable things that we can't know whether are unknowable (at least if you believe the Church-Turing thesis, which is that knowledge is computable). This may be one of them, we don't know.