Let's consider 2.1 first. You said that **married smokers** < **married people**. That is always true (unless every married person smokes, in which case the two would be equal), because anyone who is a "married smoker" is also a "married person". But since it is always true, it doesn't really give any useful information.
However, in the actual equation (2), the denominators are different. The left side is dividing by **all smokers**, and the right side is dividing by **all people**. Since not everyone smokes, those are different. Equation 2.1 drops the two denominators, which is a wrong step because the denominators are not equal.
Equation 1.1 seems to have a different error; it's trying to compare marriage and smoking to each other. We aren't given anything about this. It's possible, for instance, that 50% are married, 10% of married people smoke, and 30% of non-married people smoke. Then **married** > **smokers**, but now **married smokers** / **married people** is 10%, and **all smokers** / **all people** is 20% (check this yourself).
I wonder if you meant to write **married smokers** < **all smokers**, which is the same kind of error as for equation 2.1 if you switch the two categories.
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For trying to intuit the equations, I personally think percentages are easiest. But if you find comparing numbers of people easier, I might suggest using **married smokers < married \* (smokers / all people)**. You can think of the right side as the "naive guess" of how many married people are smokers, if you used the fraction of all people who smoke. The inequality, then, says that the actual number of married smokers is less than the number who would smoke if the two categories were independent.
thelast19digitsofpi
·
2021-06-05T14:49:27Z (almost 3 years ago)