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Consider a dart board that is represented by a unit circle centred at the origin. Each dart lands at a singular point within the circle (or on its outer edges). Arguments that the probability of the dart landing at a single point is $0$ are often unconvincing to me. It is clear that the probability must be strictly less than any positive real number: the probability cannot be $0.01$, for instance, as there are more than $100$ points on the dartboard, and so the sum of the probabilities would be greater than $1$. However, I still find this explanation unsatisfying, as $0 \times \infty$ is undefined (it does not equal $1$). Moreover, there is something deeply counter-intuitive about an event having a probability of $0$. When I see an event with probability $\frac{1}{5}$, I know that it means that if I repeat the experiment $5$ times, on average the number of successful trials should equal $1$. If something had a probability of $0$, then because $0$ multiplied by any number is still $0$, then no matter how many times I repeat the experiment, a successful trial would occur $0$ times on average. This seems very close to my conception of 'impossible', but perhaps because this is only an average, there is some nuance here which I am missing.

Thanks for reading.

Joe
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    There’s likely a tonne of related answers on MathsSE to your question. See, for example, https://math.stackexchange.com/questions/41107/zero-probability-and-impossibility – Adam Rubinson Apr 12 '20 at 13:05
  • The fact that $101\times0.01>1$ has nothing to do with $\infty$. Do you agree that the probability is less than $0.01$? If so then you should also agree that it is less than every $\epsilon>0$ (and consequently must be $0$). This is proved for every $\epsilon>0$ separately without any interference of $\infty$. – drhab Apr 12 '20 at 13:27
  • @drhab Thanks for responding. Unfortunately, the fact that it must be strictly less than any positive real still doesn't fully answer the question for me. How can the problems I highlighted about the probability equalling $0$ be resolved? For example, do the probabilities still need to add up to $1$ in an infinite sample spac?. If they do, then how so given $0 \times \infty$ is undefined. And if they don't, then why is this the case? – Joe Apr 12 '20 at 13:52
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    "Probability $0$" is not equivalent to "intuitively impossible." There's no way around that if we want probabilities to be real numbers and be able to model continuous phenomena like throwing darts. – Ned Apr 12 '20 at 14:11
  • The probability of the dart landing at a single point is not 0, but the limit with dart landing measurement precision going up or the target getting smaller is 0. Intuitively the chance of hitting a point is the same as finding a dart or drawing a target with diameter 0. And 0 times anything is 0, unless the 0 is a result of a limit, then instead of $0\infty$ we have $\frac{1}{\infty}\infty$ and the result depends on the cardinality of the sets, and thus is undefined for the general expression, but don't quote me on this. – Ymh Apr 12 '20 at 14:50
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    "Do the probabilities still need to add up to $1$ in an infinite sample space?" You have an earlier problem: what does it even mean to add up uncountably many terms? Usually in math, and for good practical reasons in probability theory, we restrict infinite summation to the countable case where the terms can be indexed by natural numbers. – Mark S. Apr 12 '20 at 15:58
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    Rather appropriately, right around the time of my previous comment popular math Youtuber Grant Sanderson, aka 3Blue1Brown just posted a Youtube video titled "Why 'probability of 0' does not mean 'impossible'", which I bet might be helpful, at least a bit. – Mark S. Apr 12 '20 at 17:32
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    @Mark S. Thanks for the help! I’ll be sure to check out that video. – Joe Apr 12 '20 at 18:04
  • How can a point with zero area exist? – Jack M Apr 12 '20 at 20:04
  • @Jack M The question I was posing was more of a mathematical abstraction rather than a true model of the real world. Of course, there would be no way for a dart to land on a single point unless it were ‘infinitely thin’, if that even makes sense. – Joe Apr 13 '20 at 12:36

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Tl;dr: I think your question is a good one- so much so, that it will take a lot of learning about the theory behind probability before you yourself will be somewhat satisfied with an answer. (Or perhaps this is me projecting my lack of knowledge in this area).

To be honest, at first I wrote a long answer, but then I decided I don’t understand all the definitions of the terms when it comes to probability (density) functions, and I know next to nothing when it comes to The Lebesgue Theory.

However, I think (although I could be wrong) that these two links are a good start to try to improve one’s understanding of what’s going on:

https://en.m.wikipedia.org/wiki/Probability_density_function (The first few paragraphs in particular)

The sum of an uncountable number of positive numbers

However, do not take this as a full answer because I reckon there is a lot more (mathematically) to OP’s question than meets the eye- probably something to do with “probability measures / measurable functions” and “Borel Sets” (basically the Lebesgue Theory) and what have you- and I know next to nothing about these, so please someone else give a proper answer. But I can see that these things are explored towards the end of Rudin’s PMA - I just haven’t got there yet. But I thought I should share my 2 cents anyway with the above links.

Adam Rubinson
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