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It wouldn't make sense if you were talking about known probabilities, e.g. with fair coin the probability of throwing heads is 0.5 by the laws of physics. However, unless you are talking about textbook example, the exact probability is never known, we only know it approximately.

The different story is when you estimate the probabilities from the data, e.g. you observed 13 winning tickets among the 12563 tickets you bought, so from this data you estimate the probability to be 13/12563. This is something you estimated from the sample, so it is uncertain, because with different sample you could observe different value. The uncertainty estimate is not about the probability, but around the estimate of it.

Another example would be when the probability is not fixed, but depends on other factors. Say that we are talking about probability of dying in car accident. We can consider "global" probability, single value that is marginalized over all the factors that directly and indirectly lead to car accidents. On another hand, you can consider how the probabilities vary among the population given the risk factors.

You can find many more examples where probabilities themselves are considered as random variables, so they vary rather then being fixed.

A most relevant illustration from xkcd:

with associated caption:

...an effect size of 1.68 (95% CI: 1.56 (95% CI: 1.52 (95% CI: 1.504
(95% CI: 1.494 (95% CI: 1.488 (95% CI: 1.485 (95% CI: 1.482 (95% CI:
1.481 (95% CI: 1.4799 (95% CI: 1.4791 (95% CI: 1.4784...

I know of two interpretations. The first was said by Tim: We have observed $X$ successes out of $Y$ trials, so if we believe the trials were i.i.d. we can estimate the probability of the process at $X/Y$ with some error bars, e.g. of order $1/sqrt{Y}$.

The second involves "higher-order probabilities" or uncertainties about a generating process. For example, say I have a coin in my hand manufactured by a crafter gambler, who with $0.5$ probability made a 60%-heads coin, and with $0.5$ probability made a 40%-heads coin. My best guess is a 50% chance that the coin comes up heads, but with big error bars: the "true" chance is either 40% or 60%.

In other words, you can imagine running the experiment a billion times and taking the fraction of successes $X/Y$ (actually the limiting fraction). It makes sense, at least from a Bayesian perspective, to give e.g. a 95% confidence interval around that number. In the above example, given current knowledge, this is $[0.4,0.6]$. For a real coin, maybe it is $[0.47,0.53]$ or something. For more, see:

Do We Need Higher-Order Probabilities and, If So, What Do They Mean?
Judea Pearl. UAI 1987. https://arxiv.org/abs/1304.2716

All measurements are uncertain.

Therefore, any measurement of probability is also uncertain.

This uncertainty on the measurement of probability can be visually represented with an uncertainty bar. Note that uncertainty bars are often referred to as error bars. This is incorrect or at least misleading, because it shows uncertainty and not error (the error is the difference between the measurement and the unknown truth, so the error is unknown; the uncertainty is a measure of the width of the probability density after taking the measurement).

A related topic is meta-uncertainty. Uncertainty describes the width of an a posteriori probability distribution function, and in case of a Type A uncertainty (uncertainty estimated by repeated measurements), there is inevitable an uncertainty on the uncertainty; metrologists have told me that metrological practice dictates to expand the uncertainty in this case (IIRC, if uncertainty is estimated by the standard deviation of N repeated measurements, one should multiply the resulting standard deviation by $frac{N}{N-2}$), which is essentially a meta-uncertainty.

Error bars are shorthand for a probability distribution. So error bars on errors bars are shorthand for a probability distribution for a probability distribution. That entails assigning probabilities to probabilities, so this is what we need to study.

Following John Maynard Keynes, a probability is the degree of belief that we are justified in entertaining in one proposition, in relation to another. Now there are many pairs of intelligible propositions for which we don't know how to assign a numerical probability. But for my part, I cannot think of a pair of plain propositions (not themselves containing a probability phrase) which yield a numerically quantified uncertain probability. If we try to generate one, it always entails an ordinary probability. For example, suppose that our given proposition is
$$
mathcal{I} = unicode{8220} text{die 1 has 6 sides, and die 2 has 7 sides; one of these dice is rolled} unicode{8221}
$$
and that the proposition we're assessing against $mathcal{I}$ is
$$
mathcal{A} = unicode{8220} text{the number showing is 3} unicode{8221}
$$
Then it's tempting to set
$$
mathrm{prob}left(mathrm{prob}(mathcal{A} | mathcal{I}) = p | mathcal{I}right) = begin{cases}
frac{1}{2} & p = frac{1}{6} \
frac{1}{2} & p = frac{1}{7}
end{cases}
$$
which attempts to assign a probability distribution to a probability. Quite apart from the confusion introduced by the nested probability phrases, this cannot be correct, because $mathrm{prob}(mathcal{A} | mathcal{I})$ is deducible and therefore not subject to uncertainty. Let
$$
mathcal{D}(d) iff unicode{8220} text{die $d$ is rolled} unicode{8221}
$$
Then
begin{align}
mathrm{prob}(mathcal{A} | mathcal{I}) &= sum_d mathrm{prob}(mathcal{A}, mathcal{D}(d) | mathcal{I}) \
&= sum_d mathrm{prob}(mathcal{A} | mathcal{D}(d), mathcal{I}) : mathrm{prob}(mathcal{D}(d) | mathcal{I}) \
&= sum_d mathrm{prob}(mathcal{A} | mathcal{D}(d), mathcal{I}) cdot frac{1}{2} \
&= frac{1}{2} left( frac{1}{6} + frac{1}{7} right) \
&= frac{13}{84}
end{align}

Nested probabilities, in short, cannot be sustained, and neither, therefore, can nested error bars.

There are very often occasions where you want to have a probability of a probability. Say for instance you worked in food safety and used a survival analysis model to estimate the probability that botulinum spores would germinate (and thus produce the deadly toxin) as a function of the food preparation steps (i.e. cooking) and incubation time/temperature (c.f. paper). Food producers may then want to use that model to set safe "use-by" dates so that consumer's risk of botulism is appropriately small. However, the model is fit to a finite training sample, so rather than picking a use-by date for which the probability of germination is less than, say 0.001, you might want to choose an earlier date for which (given the modelling assumptions) you could be 95% sure the probability of germination is less than 0.001. This seems a fairly natural thing to do in a Bayesian setting.