Let us imagine a study where A+C number of people with a certain disease were compared with B+D number of people without the disease. We would like to find an association of Exposure with this disease. We see that out of A+C number of people with the disease, A number of people were exposed. So the probability of exposure for people with the disease was A/(A+C), and the probability of non-exposure was C/(A+C). If we put these two numbers together, we can see that the odds of Exposure for people with the disease was A:C. Similarly, if we follow the same logic of exposure for people who did not have the disease ("without disease"), we can see that their odds of exposure would be B:D. The odds ratio (OR) as a measure of the association between the exposure and the disease would be given by A:C / B:D, and would be equivalent of A * D / B * C or the cross product ratio.
We now apply this logic to the following table of Radon Gas exposure and Lung Cancer: