Using probability to determine dependent events In my HS math class, I'm teaching Prob & Stats.  We are studying how the intersection of events can show dependence.  For example, we took a survey of students in the HS:  57% of those surveyed are in Extra-curricular Activities, 65% of those surveyed are on the Honor Roll, and 46% are both.  If these two events were independent, the the "both" (intersection) would be equal to their product: P(EC) * P(HR) = 37%.  Since the observed intersection is higher, these have a positive correlation.
Is this valid statistics?  I.e., is this kind of data collection and statistical analysis something that is done in research, in order to show events are independent or dependent?  I'd like to be able to tell my students that "This is how it is done in the 'Real World.'"  Can anyone give me examples of real-world situations where this is done?
 A: The chi-square test uses the idea you mentioned to test for independence of variables. The null hypothesis in the chi-square is that the data are independent and it constructs the numbers one should expect to see along the same lines of what you mentioned in your question. The expected frequencies are then compared with observed to assess if the variables are in fact independent of each other. 
A: In the real world you would need to be a bit careful with just saying the size of the intersection is a bit large therefore there's a correlation.
You would need to work out what the probability that the intersection were higher than the one you observed if they were independent)
(That is if you assume each pupil is has a $35\%$ chance of being on both the honour roll and doing ECA (the null hypothesis that the events are independent)  what is the probability that 46% of your survey were both the honour roll and doing ECA.  This depends on the size of your survey. (a small survey might produce the result by chance)
There are three absolutely classic examples.  The first is the link between smoking and lung cancer.  which was shown in exactly the way you argued.   There's also Florence Nightingale who used similar ideas to demonstrate a link between various diseases and conditions in military hospitals during the Crimean war, another great one is John Snow who proved that cholera was a waterborne disease by showing that people who used a particular street pump were vastly more likely to get cholera.  Both these last two examples lead to huge improvements in sanitation around the world.
