 Bandyopadhyay et al. (2011) reject Pearl’s causal analysis of the paradox, and defend an alternative mathematical explanation. They note that there can be instances of the paradox that do not seem to invoke any causal notions. For example, suppose we take the proportions in Table 1 not to refer to the proportions of recovering/non-recovering patients among the treatment/non-treatment groups in male and female populations, but rather to the proportions of red and blue marbles among big or small marbles in two bags. Suppose that in either bag the big marbles have a higher red-to-blue ratio than the small marbles. Bandyopadhyay et al. plausibly claim that in this case, it would be surprising to discover that, were we to pour the bags into a single box, the small marbles have a higher red-to-blue ratio than the big ones. If there are cases of the paradox that still exhibit surprise despite having nothing to do with causality, then the general explanation of the paradox cannot be causal.[9]