In caring for elderly patients at a mental hospital, a major goal of the hospital is successful placement of the patients in boarding homes or nursing homes.

In caring for elderly patients at a mental hospital, a major goal of the hospital is successful placement of the patients in boarding homes or nursing homes. The movement of patients between the hospital, outside homes, and the absorbing state (death) may be described by the following Markov chain (the unit of time is one month):

Each month that a patient spends in the hospital costs the state $655, and each month that a patient spends in a home costs the state $226. To improve the success rate of the placement of patients in homes, the state has recently begun a “geriatric resocialization program” (GRP) to prepare the patients for functioning in the homes. Some patients are placed in the GRP and then released to homes. These patients presumably are less likely to fail to adjust in the homes. Other patients continue to go directly from the hospital to homes without taking part in the GRP. The state pays $680 for each month that a patient spends in the GRP. The movement of the patients through various states is governed by the following Markov chain:

a Does the GRP save the state money?

b Under the old system and under the GRP, compute the expected number of months that a patient spends in the hospital.