Answer: E

Small Hint:

List all positive divisors of $60$

Big Hint:

Count how many of the $12$ divisors are less than $7$

Solution:

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.