junior college 2 chevron_right H3 Maths

I have been struggling to answer this question, specifically how to incoporate the k or more 4s immediately followed by a 5 or 6 aspect. What I've tried so far is:
for k=0
P[homer] = (1/3) + (1/3)(2/3)^2 + (1/3)(2/3)^4 + ....
P[marge] = (1/3)(2/3)^1 + (1/3)(2/3)^3 + (1/3)(2/3)^5 + ....

P[marge] = 2/3 * P[homer]
P[marge] + P[homer] = 1
5/3 P[homer] = 1
P[homer] = 3/5

I'm not sure if thats right, and I'm also not sure what to do for the other cases of k. If anyone could help me solve this problem it would be much appreciated.