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To test two probabilities for independence one way is to check whether $P(A, B) = P(A) \cdot P(B)$.

However, in a practical setup, the probabilities may have some confidence intervals. So, $P(A)$ is actually $P(A) \pm c_A$, $P(B)$ is actually $P(B) \pm c_B$ and $P(A,B)$ is actually $P(A,B) \pm c_{AB}$ (95% confidence for all variables).

Is there a standard way to assess for independence by taking into account the confidence intervals?

Also, does it make sense to "quantify" the independence by using some metric, for example the overlap between confidence intervals. In my application I measure the two variables day by day, and what I observe is that they start by being strongly dependent, but they become independent as time goes by.

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    $\begingroup$ To check whether two random variables $X$ and $Y$ are independent one has to check that $P(X\in A, Y\in B)=P(X\in A)P(Y \in B)$ for all $A$ and $B$. $\endgroup$ – Stéphane Laurent Oct 7 '14 at 7:12
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I would to this: 1. for each actual osservation create two binary variables: happened_a, happened_b; 2. collect a sample 3. Calculate any statistic based on crosstabulations: chi square, or, much bette, odds ratio. For these two statistics, formulas to compute confidence intervals for a given confidence level $\alpha$ are available also in closed form. 4. If you had chosen, for example, the odds ratio, the two variable would have been defined independent if the odds ratio's confidence interval would not comprehed the one.

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