Chi^2 test for 2x2-contingency table using probabilities instead of count data

Question

I would like to perform a $\chi^2$ hypotheses test for association based on a $2 \times 2$ contingency table. However, my contingency table

$\begin{tabular}{|c|c|} a & b \\ c & d \end{tabular}$

does not contain integer count data but probabilities

$\begin{tabular}{|c|c|} 1 - f(a) & 1 - g(a) \\ f(a) & g(a) \end{tabular}$

$f,g$ are Bernoulli distributions over event $a.$ How can I calculate the test statistic in this case?

Answer 1

You can't, unless you know the sample size. A $\chi^2$ test is a test of whether observed data are plausibly generated from an underlying population with no association between the two variables. Unless you know how many data were actually observed you can't make any conclusions about whether they are plausibly from that null-hypothesis distribution.

If you do know the total sample size, obviously you can just convert your probabilities to counts and perform the test the usual way.