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I am trying to understand $P(A|B)$ in practice. I know that $(A,B)$ is the intersection between $A$ and $B$ on the diagram, and I know as well that $P(A|B)$ is probability of $A$ given $B$. But where is $P(A|B)$ on the diagram ?

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    $\begingroup$ It is not on the diagram. $\endgroup$ – Xi'an Jan 2 '19 at 10:30
  • $\begingroup$ How can I represent it (on my mind) ? $\endgroup$ – Mehdi Souregi Jan 2 '19 at 10:32
  • $\begingroup$ Is it a part of (A,B) or part of A, I am confused $\endgroup$ – Mehdi Souregi Jan 2 '19 at 10:34
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Using the definition of $P(A|B)$ which is equal to $\frac{P(A\cap B)}{P(B)}$. Hence, you can't show it explicitly on the diagram as it is defined base on the division of two parts on the diagram. The value of blue part over the value of red circle. Hence it is $\frac{0.1}{0.3 + 0.1} = \frac{1}{4}$.

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Remember that

$$ P(A|B) = \frac{P(A, B)}{P(B)} = \frac{0.1}{0.3 + 0.1} = \frac{1}{4}, $$

which means that $P(A|B)$ is given by the proportion of the blue zone in your picture with respect to the red $B$ circle. This is not immediately visible in the diagram, so you'll have to use your imagination a bit to see the blue zone being $1/4$ of the size of the red circle.

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