The funny coin will be picked 1/4 of the time. A fair coin will be chosen 3/4 of the time and come up heads twice 1/4 of the time, which means this sequence occurs 3/4 * 1/4 = 3/16 of the time. Thus, HH will appear 3/16 + 1/4 = 7/16. So the probability that HH is caused by the funny coin is 1/4 / 7/16 = 4/7.