Are there any examples of events A and B such that P(A∩B|C)=P(A|C)P(B|C) but not with C'? Especially with dice or coins? or is there any specific example?
2026-04-07 03:07:47.1775531267
Give an example in which two events exhibit conditional independence with an event B but not with B complement.
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There is an urn with two balls, black and white. You also have a fair coin. You flip the coin, and then...
If the result is heads, you draw two balls from the urn with replacement.
If the result is tails, you draw two balls form the urn without replacement.
Let $C$ be the event the coin is heads, let $A$ be the event the first ball you draw is white, and let $B$ be the event that the second balls is black. Clearly, $A$ and $B$ are conditionally independent given $C$, but $A$ and $B$ are conditionally dependent given $\overline C$.