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For example, what is the probability that John will land a coin on heads two times in a row on Tuesday if he only gets to do two tosses? Clearly it’s 1/4.

Now, what is the probability that we will ever have a rock form naturally be shaped exactly in the face of Abraham Lincoln?

Now, the latter seems much more incredibly improbable. But what if we had trillions of years for this? Since we want the probability of the second event happening at any time, could it still be higher than 1/4?

Intuitively, it seems that it should still be lower unless we literally have an infinite amount of time but it’s hard to get my head around this.

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    You might also like: Cydonia
    – Hudjefa
    Commented Mar 25 at 9:04

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The two examples, coin and Lincoln rock, use two different concepts of probability.

  • The coin example knows the probability 1/2 for the head of a fair coin. Accordingly it computes the probability of two heads in sequence as 1/4 = (1/2) x (1/2).
  • The Lincoln example does not know the probability that an arbitrary rock is a Lincoln rock. Not only that one does not know the probability. The probability is not well-defined. One speaks only informally about the probability of a Lincoln rock.
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  • I liketh your answer sir!
    – Hudjefa
    Commented Mar 25 at 9:00
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    "The probability is not well-defined." Moreover, the property of 'looking like' Abe Lincoln is not well defined.
    – JimmyJames
    Commented Mar 25 at 16:12
  • There is a difference between "not well-defined" and "hard to calculate". Rocks do take random shapes, due to erosion and other factors, and there is always a chance that a rock will take the shape of Lincoln's face.
    – Stef
    Commented Mar 26 at 10:18
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    @AgentSmith If thou wilt use an archaic form, at least conjugate it correctly :-)  (“I like”, “thou likest”, “he/she/it liketh”, “we like”, “ye/you like”, “they like”.)
    – gidds
    Commented Mar 26 at 12:38
  • @gidds, English as she is spoke (my reference book)
    – Hudjefa
    Commented Mar 26 at 15:53
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The game can only be won or lost.

The probability of winning is X.

The probability of losing is the probability of not-winning, which is 1-X.

We want to know how likely it is that we win the game at least once out of N games.

The only way to win at least one game out of N games is to not [lose N games out of N games]

= 1-[lose N games out of N games].

= 1-[lose the first game][lose the second game][lose the third game]...[lose the Nth game]

= 1-(1-X)^N

Whether spontaneous statuary is a good example of such a game, I leave up to the spontaneous statuarians. But 1-(1-X)^N is a good formula for everyone to memorize. There are many things in life, business, and games of chance for which the trials are more or less statistically independent; and in which we care the chance of winning/losing at least once out of two or more trials, not the chance of winning/losing in a single trial.

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The excellent answer provided by GS gives you the formula for how the probability of an event increases over time if you know the probability of it occurring in unit time. The formula makes it clear that the probability approaches 1 as the number N approaches infinity, regardless of how small is the probability of the event happening in unit time. However, that formula assumes the probability of the event occurring in unit time is constant.

With many natural continuous events, probabilities do not remain constant. For example, you might consider the probability that I could run a mile in four minutes. By applying the formula given by GS, you might suppose that if I failed miserably to meet the target, I will nonetheless meet it ultimately if I don't stop running. However, in reality my times will get steadily worse, not better, as I exhaust and injure myself.

To take your rock weathering example, the mechanisms that cause weathering might tend to apply symmetrically, so a rock with a characterful shape might simply become more and more rounded over time. The probability of it becoming Lincoln-like might be steadily reducing with time, and the rock might wear away to nothing long before infinite time passes.

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  • Very interesting points! I didn’t think of that assumption.
    – Baby_philosopher
    Commented Mar 25 at 10:36
  • Probabilities can be expressed as functions of time and of other variables, so we can speak of the probability of running the nth. consecutive mile in under four minutes, or of doing so from a rested state as a function of one's age.
    – sdenham
    Commented Mar 25 at 18:24
  • ...and, probably even more paradoxically than @Baby_philosopher 's example, they can remain positive forever and still not amount to certainty. One example is an event with a probability of 1/4 over the first interval, and which halves with each successive interval, converging on one-half
    – sdenham
    Commented Mar 25 at 18:52
  • Very interesting points indeed! Probabilities of something happening can tend to 0, or 1, ...... and under some assumptions it's often the case that they cannot tend towards any other intermediate value!! Low-probability events, particularly as the number of trials approaches infinity?
    – Stef
    Commented Mar 26 at 10:23
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Now, what is the probability that we will ever have a rock form naturally be shaped exactly in the face of Abraham Lincoln?

Good question. If n is the number of rocks that look like Abe and N is the number of rocks on Earth. Then the probability will be n/N.

If no rock has been found that looks like Abe, then the initial probability is 0 until one rock is found that looks exactly like Abe. The difficulty is keeping a log/count of every rock examined while performing this search.

For an accurate probability, all the rocks on Earth must be examined.

To compare this to a sequence of coin tosses, the ordered sequence of tosses must be as complex as the ordered sequence of cuts, chisels, abrasion, etc required to create Abe's face:

What is the probability that the outcome of N consecutive coin tosses will spell out in binary. "It's Abe's face!" Where N is the binary length of the phrase.

In simple 2D plotting graphics, Abes face could be completely described as the sum of sequential turtle graphics commands. This can be used to determine the probability of Abe's face showing up as a function of random turtle graphic commands.

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    You would also need to define what qualifies as a rock and what qualifies as looking like Abe Lincoln.
    – JimmyJames
    Commented Mar 25 at 16:09
  • @JimmyJames Yes. Grains of sand could be considered tiny rocks. I don't know how facial recognition software works but those parameters are a good start for qualifying what Abe looks like.
    – Idiosyncratic Soul
    Commented Mar 25 at 16:21
  • The other parameter involved, since the OP mentioned time, is an estimate of how long it takes for a rock to substantially change it's configuration such that it would constitute an independent trial.
    – Dave
    Commented Mar 25 at 23:05
  • I would call n/N an estimate of the probability that a given rock will naturally take the shape of Lincoln. If no rock has been found yet, then the estimate is 0, but that doesn't mean the probability it 0.
    – Stef
    Commented Mar 26 at 10:29

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