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Three events $A, B, C$ each occur with the same probability $p$, which takes any value between 0 and 1.

Find the smallest and largest possible values for the probability of the union $P(A\cup B\cup C)$, if they are not know to be independent (so these events can either be independent or dependent).

The largest possible value, I think would be $p+p+p$, assuming that these events are mutually exclusive. However, I am not sure how to find the lower bound on this probability. I have a hunch that I could use this formula:

$$\begin{array}{rcl} P(A \cup B \cup C) &=& P(A) + P(B) + P(C)\\ && - P(A \cap B) - P(A \cap C) - P(B \cap C)\\ && + P(A \cap B \cap C) \end{array}$$ assuming that the events are independent.

So is it safe to say that the probability is maximized when the events are mutually exclusive, and minimized when the events are independent?

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The number $p$ is fixed. The union is "smallest" if $A=B=C$, in which case the probability of the union is $p$.

To make the probability of the union "big," separate $A,B,C$ as much as possible. As you saw, the answer is sort of $3p$. More precisely, it is $3p$ if $3p\le 1$, that is, $p\le 1/3$, and $1$ if $p\gt 1/3$.

"Formulas" are not of great help in this problem, a Venn diagram is more informative. If the "area" of each of $A,B,C$ is $p$, then the union has smallest area if the three sets are the same.

To make the combined area biggest, we try to arrange for $A,B,C$ to overlap as little as possible. If $p\lt 1/3$, we can set up situations in which there is no overlap at all, in which case the combined area is $3p$.

In general, both in the minimum case and the maximum case, we have dependence.

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  • $\begingroup$ I don't quite understand why the probability of the union is $p$ if $A=B=C$. $\endgroup$ – Luchia Dec 7 '13 at 21:33
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    $\begingroup$ You will be happy if you land in the union of $A, B, C$. Your probability of landing in $A$ is $p$, so your probability of happiness is at least $p$. If we make $B=C=A$, then your chance of happiness is $p$, since then $A\cup B\cup C=A$. $\endgroup$ – André Nicolas Dec 7 '13 at 21:39

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