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I was looking at a reference today and saw a conditional probability written with more than 1 vertical bar $P(A|B|C)$. This is the first time I've seen something like this. Is this valid syntax. Is there another way to represent this?

user5965026
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  • What reference? – Angina Seng Apr 18 '20 at 14:10
  • @AnginaSeng. It's from some handwritten lecture notes that I have. – user5965026 Apr 18 '20 at 14:11
  • @AnginaSeng I feel like based on context, it seems to be $P(A|B,C)$. This was regarding a 2 player game russian roulette game with a single bullet located in 6 chambers. I saw them write P(A|B), where A is the event that the first player survives the second round given that the first player survives the first round (represented with B). C is the event that the 2nd player survives the first round. Then they wrote $P(A|B) = P(A|B|\bar{C})P(\bar{C}) + P(A|B|C)P(C)$ i.e., the total probability formula. – user5965026 Apr 18 '20 at 14:15
  • In Yor Revus book, if I'm not mistaken, they sometimes write $\mathbb E(X\mid Y\mid Z)$ for $\mathbb E[\mathbb E[X\mid Y]\mid Z]$. – Surb Apr 18 '20 at 14:16
  • @Surb Is that just for expectation, or did write something for probability notation too? I feel like in my case, the person meant to write $P(X|Y \cap Z)$ – user5965026 Apr 18 '20 at 14:18
  • @user5965026: You can adapt as follow : $$\mathbb P(X\in A\mid Y\mid Z)=\mathbb E[\mathbb E[\boldsymbol 1_{{X\in A}}\mid Y]\mid Z]=\mathbb E[\mathbb P(X\in A\mid Y)\mid Z].$$

    But in your precise case, I don't know. They should have defined it somewhere. The notation $\mathbb P(A\mid B\mid C)$ for $\mathbb P(A\mid B\cap C)$ seems a bit silly at my point of view. But it could means that. I really don't know.

     – Surb Apr 18 '20 at 14:36

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