2003 AMC 10A Problem 8

The factors of $60$ are $1,$ $2,$ $3,$ $4,$ $5,$ $6,$ $10,$ $12,$ $15,$ $20,$ $30,$ and $60.$ Six of these twelve factors are less than $7,$ so the probability is $\dfrac{6}{12} = \dfrac{1}{2}.$ Thus, the correct answer is E.