Every day a professor leaves their home in the morning and walks to their office. Every evening they walk home. They take their umbrella with them only if it is raining. If it is raining and they do not have their umbrella with them (at their home or office), then they must walk in the rain. Suppose that it rains with probability 1/3 at the beginning of any given trip independently of all other trips. Show that 63/16 4 is the expected number of days until the professor must walk in the rain without their umbrella (either that morning or evening), supposing that initially they have their umbrella with them at home.

Hint: Let be the expected number of days supposing they initially have their umbrella with them at home, and let be the expected number of days supposing that they do not. Explain why

=92+95(1+)+92(1+)

and then, similarly, find an equation for in terms of and . Use these equations to solve for .