If P(A) = 0.2, P(B) = 0.2, P(C) = 0.45, P(D) = 0.15
Then does this mean that A and D are mutually exclusive?
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$\begingroup$ Please consult the definition: does it say anything at all about probabilities?? $\endgroup$– whuber ♦Sep 25, 2017 at 21:04
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1$\begingroup$ The answer to your question is No as Kodiologist has already said, but I don't think he got at the root cause: when thinking about multiple events, many beginning students assume that "Event $A$ occurred" is the same as saying that "$A$ **and only **$A$ occurred; all other events did not occur" and so with $P(A)+P(B)+P(C)+P(D) = 1$, it is natural to assume that this is an indication that the events are mutually exclusive. Not so. "$A$ occurred" tells us only that neither $A^c$ nor any subset of $A^c$ occurred. As to the occurrence of any other event, the occurrence of $A$ tells us nothing. $\endgroup$– Dilip SarwateSep 26, 2017 at 2:23
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No. For example, let $X$ be uniformly distributed on $[0, 1]$ and let $A$ be the event that $X ≤ .2$, $B$ be equal to $A$, $C$ be the event that $X ≤ .45$, and $D$ be the event that $X ≤ .15$. All the hypotheses are satisfied, but far from $A$ and $D$ being mutually exclusive, $P(D|A) = 1$.
answered Sep 25, 2017 at 21:04