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I am new to stats and I have a question regarding the following statement: If Pr(A) = 0, it doesn’t mean that event A is an empty set.

However, isn’t the probability of A = (number of sample points in A) / (number of sample points in the same space) and that events are subset of the sample space.

Thus, based on the formula above, the only way the Probability of A is 0 is if the number of sample points in A is 0 thus making it an empty set. Furthermore, A cannot contain elements that are outside the sample space thus A would be an empty set.

However, this is contradictory to the statement If Pr(A) = 0, it doesn’t mean that event A is an empty set.

Pitouille
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Mememaster
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    See https://stats.stackexchange.com/questions/142730/px-x-0-when-x-is-continuous-variable – Tim Sep 03 '21 at 10:31
  • see tim's link. what happens to your formula if there are an infinite number of points (eg uniform distribution over real numbers in [0,1]) – seanv507 Sep 03 '21 at 11:22

2 Answers2

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The rule that $\mathbb{P}(A) = |A|/|\Omega|$ (for sets $A\subseteq \Omega$) arises in a specific probability context, where you have a sample space $\Omega$ containing a finite number of outcomes all with equal probability. In that particular context, you you are correct that $\mathbb{P}(A)=0$ only when the set $A$ is empty. However, probability theory deals with much more general situations than this, so that rule holds only in a very narrow class of cases.

More generally, probability theory deals with cases where there are events $A$ that are non-empty sets of outcomes, but they still have zero probability. This leads us to make a distinction between events that are certain to occur and events that are merely almost sure (i.e., occur with probability one).

Ben
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$A$ doesn't have to be an empty set. For instance, consider all living people queuing in a straight line. So $S = \{ \text{ all living people } \}$. Let $B= \text{ You are at the front of the line } $.

Now $P(B)\approx 1/7\text{billion} = 1.4 \times 10^{-10}$. Now consider the set $T = \{ \text{ all dead people } \}$. Note that $S \cap T = \emptyset$ (the intersection of $S$ and $T$ is empty). The probability a dead person (say, $A = \text{Genghis Khan}$) is as the front of the line is $0$. This is because $A \cap S = \emptyset$.

In short, $P(A) = 0$ doesn't imply $A = \emptyset$ but rather $A \cap S=\emptyset$.

jcken
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