 Consider, for example, whether a student at University X having high grades (\(E\)) says anything about the likelihood of her taking a class in philosophy (\(H\)). The registrar tells us that 35% of students take a philosophy class at some point, so \(p(H) = 35/100\). They also tell us that only 20% of students campus-wide have high grades (defined as a GPA of 3.5 or above), so \(p(E) = 20/100\). But they don’t keep track of any more detailed information. Luckily, the philosophy department can tell us that 25% of students who take their classes have high grades, so \(p(E\mid H) = 25/100\). That’s everything we need to apply Bayes’ theorem:  \[\begin{split} p(H\mid E) &amp;= p(H)\frac{p(E\mid H)}{p(E)}\\ &amp;= 35/100 \times \frac{25/100}{20/100}\\ &amp;= 7/16\end{split}\]