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?


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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