AMC 10 2008a Problem 22

Jacob uses the following procedure to write down a sequence of numbers. First he chooses the first term to be 6. To generate each succeeding term, he flips a fair coin. If it comes up heads, he doubles the previous term and subtracts 1. If it comes up tails, he takes half of the previous term and subtracts 1. What is the probability that the fourth term in Jacob's sequence is an integer? $\mathrm{(A)}\ \frac{1}{6} \mathrm{(B)}\ \frac{1}{3} \mathrm{(C)}\ \frac{1}{2} \mathrm{(D)}\ \frac{5}{8} \mathrm{(E)}\ \frac{3}{4}$