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How would I find P ((A|B)|C)? Do I substitute the formula for P (A|B), P=(A ∩ B))/(P (B)), and then redo the function to get (P (A ∩ B ∩ C))/(P (C)^2)?

I heard that P=(A|B ∩ C) can be used, but why would that be equivalent, unless you assume that C is independent of A? Given a three-circle Venn diagram, the initial sample size would be too restricted.

Katie Tibbs
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2 Answers2

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The notation $\mathsf P((A\mid B)\mid C)$ is not standard.   There should only be one bar between the event being measured and the condition.   When conditioning over two events, take the conjunction.

Both $\mathsf P(A\mid B, C)$ and $\mathsf P(A\mid B\cap C)$ mean the conditional probability of $A$ given $B$ and $C$.

$$\mathsf P(A\mid B\cap C) = \frac{\mathsf P(A\cap B\mid C)}{\mathsf P(B\mid C)} = \frac {\mathsf P(A\cap B \cap C)}{\mathsf P(B\cap C)}$$

Graham Kemp
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Yes, $P((A|B)|C)=P(A|B\cap C)$ In both cases you are given that both $B$ and $C$ happened and asked for the probability of $A$ given that information.

Ross Millikan
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  • Can you kindly show how this relation holds? – q126y Jan 15 '19 at 07:48
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    @q126y: The Venn diagram argument is that the $|C$ restricts us to the area inside the $C$ circle. The $|B$ restricts us to the $B\cap C$ lnes. Then $P(A)$ under this assumption is the central rouleaux divided by the lens. – Ross Millikan Jan 15 '19 at 16:21