I'm an undergraduate student in pure mathematics, and I'm taking a probability theory course based on the book A First Course in Probability by Sheldon Ross. My lecturer defined the conditional probability $P(A|B)$ of two events $A,B$ by $P(A|B)=\frac{P(A\cap B)}{P(B)}$, provided that $B\neq\emptyset$. I knew that $P(\emptyset)=0$, and after some research, I realized that the converse is not true, namely, from $P(E)=0$ we can't conclude that $E=\emptyset$. I wrote to my lecturer, and among other things, I gave him a simple counterexample. (First I asked him to prove his claim, but instead he "proved" to me that $P(\emptyset)=0$, but he relied on $P(E^c)+P(E)=1$ which was proved in the book by "the finite version" of Axiom 3, which was proved using Axiom 3 and the fact that $P(\emptyset)=0$.) The simple counterexample was the sample space $S=\{1,2\}$ together with the function $P$ defined by $P(\{1\})=0,P(\{2\})=1,P(\{1,2\})=1,P(\emptyset)=0$, it's a simple matter to verify that $P$ satisfies the three axioms of probability, yet $P(\{1\})=0$ and $\{1\}\neq\emptyset$. Yet he went in circles claiming that every sample point in the sample space (i.e., a singleton subset of the sample space) needs to have positive probability—which is of course nonsense. I was getting the feeling that he doesn't like rigour, and on further researching, I found out that his background is in engineering and applied statistics. Is it normal that he has this false belief (I mean normal for people who are more interested in applied probability rather than rigorous mathematics)?
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