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I know $P(E|A)$ and $P(E|B)$, how do I know $P(E|A,B)$? Assuming $A,B$ independent.

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Welcome to MSE! Generally, it helps the MSE Community if you add your thoughts or what you have tried to figure things out and to make questions as clear as possible. Regards – Amzoti Jan 18 '13 at 0:15
up vote 3 down vote accepted

Consider this example:

  • $A$ = first coin lands Head,
  • $B$ = second coin lands Head,
  • $E_1$ = odd number of Heads,
  • $E_2$ = even number of Heads.

The first two conditional probabilities are both $1/2$. The third is $0$ for $E_1$ and $1$ for $E_2$.

(Also: Welcome to Math.SE, and please do not try to make your questions as short as possible. Adding your thoughts, even tentative, is always a plus.)

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