# When people saying 'with high probability', are there two possible interpretations?

I've seen two meaning of 'with high probability' in the literature, and I think they are different. I am not sure whether I miss understood them. Here are they:

(1) In mathematics, an event that occurs with high probability (often shortened to w.h.p. or WHP) is one whose probability depends on a certain number n and goes to 1 as n goes to infinity, i.e. the probability of the event occurring can be made as close to 1 as desired by making n big enough. (source: Wikipedia) For example here An event Π is said to occur with high probability if $$P(Π)\geq 1-\frac{c}{n^\alpha}$$.

(2) In the context of concentration, if we have $$P(X\geq a+t)\leq \exp(-t^2)$$, then we say $$X with high probability. This means that the larger the deviation from what we expected ($$X), the smaller the probability is. For example, theorem 3.1.1. proved $$P\{\|X\|_2-\sqrt{n}\geq t\}\leq 2\exp(-\frac{ct^2}{K4})$$, and call it 'with high probability $$X$$ takes values very close to the sphere of radius $$\sqrt{n}$$'. Also, I noticed that in some concentration bound, it can depend on $$n$$ as well.

From my understanding, the first one is saying the probability is bounded by a function of $$n$$, while the second one is bounded by a function of deviation, which is commonly addressed topic in the context of concentration.

I am not sure whether 'with high probability' is a vague word, and its definition depends on the context, or I misunderstood something.

Could someone help to clarify more? Thanks in advance for any comment, answer.

• If an event is not sure but has probability $1$ , we say it occurs almost surely. What "high" means depends on the context. In some cases , $p=0.8$ is already consiederd to be "high". Commented Jul 17, 2023 at 8:45
• I don't think the phrase "high probability" has a mathematical meaning. In statistics (and science) it's usually compared against the null hypothesis and the probability that the null hypothesis predicts the results, which is typically a 5-10% chance. So if an experiment "passes" this test, that there's only a 5-10% chance random results could explain it, then we say it's "highly probably"--although what we're actually saying is there's a 90-95% chance we're correct. Commented Jul 17, 2023 at 8:47
• ...and to add to that, that's why repeatability is so important in science. If I do an experiment with 90-95% accuracy, someone else should also be able to repeat that experiment with the same (or similar) results--if they don't then my experiment is flawed. The more it's repeated with that same 90-95% threshold, the more accurate it becomes. Commented Jul 17, 2023 at 8:50
• It's really just some convention by the author. Tao makes this notion exact in "Topics in Random Matrix Theory." Commented Jul 17, 2023 at 9:16

In the example you gave, the idea that they are trying to communicate is for any $$\epsilon>0$$, $$\mathbb P[||X||_2\geq (1+\epsilon)\sqrt n]\leq 2\exp\Big(-\frac{c\epsilon^2}{K4}n\Big)=e^{-\Theta(n)}=o(1)$$ Indeed for any tiny deviation $$f(n)=\omega(n^{-1/2})$$ $$\mathbb P[||X||_2\geq (1+f(n))\sqrt n]\leq 2\exp\Big(-\frac{c}{K4}\big(f(n)\sqrt n\big)^2\Big)=e^{-\omega(1)}=o(1)$$ Thus if we give the right definition of "very close", then it is precisely true that "with high probability" $$X$$ is very close to the sphere. Often the definition of "very close" comes from context, or as needed for a proof.