How to intuit : married smokers × all people < all smokers × all married people ?
How can I intuit inequality (3) in my previous post? The author intuits inequalities (1) and (2), but not (3).
Scilicet, how can you explain inequality (3) to a 10 year old? I can't intuit the meaning of multiplying smokers $\times$ people!
If you multiply both sides of each inequality by the common denominator (all people) × (all smokers) you can see that the two statements are different ways of saying the same thing:
(married smokers) × (all people) < (all smokers) × (all married people) Why doesn't
$\tag{3}$work?
In the same way, if smoking and marriage were positively correlated, it would mean that married people were more likely than average to smoke and smokers more likely than average to be married.
Ellenberg, How Not to Be Wrong (2014), page 348.
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