1. Prison Planning As the aide to the newly elected governor of your state, you have assumed respon- sibility for implementing the campaign promise to define a set of changes in the operation of the penal system. You have given some thought to the matter and have decided that you have essentially one instrument that you can control and one resource that you must provide. In the first instance, you could persuade the courts and probation boards to raise or lower the average length of incarceration. The average time in prison after each conviction is now four years, but you believe that you could move that average up or down by one year. The resource you must provide is the staff of probation officers. Each probation officer can handle an average of 25 cases at a time. Conditions in your state result in the annual conviction of about 1% of the population not in jail or on probation. All prisoners who are released from jail must enter probationary status. In any given year, 30% of the ex-convicts on probation are convicted of a new crime and returned to prison, but 10% are fully rehabilitated and return to the "general population" (that is, those not in jail or on probation); the rest remain on probation. Assume that the current policies have been in operation for a long time, and thus the distribution of the population has reached an equilibrium. It costs $50,000 per year to maintain a prisoner in jail, but only $10,000 per year (including the salary costs of probation officers) to keep a prisoner on probation. There are one million people in your state, including those in prison or on probation. (a) Draw a diagram to indicate the probabilistic flows through this system. Make sure to indicate clearly the states and annual transition probabilities. (b) What is the expected number of people who are neither in jail nor on probation? (c) What is the expected number of persons in jail? (d) How many probation officers are needed to maintain the average caseload? (e) What is the annual cost of operating the state penal system? (f) What is the annual conviction rate in the state? (g) Given what you know about the system, what changes might you recom- mend to the governor? Remember, you can control (within limits) the average prison sentence, and you also have flexibility with respect to the number of pro- bation officers. In discussing your recommended changes, make sure to provide estimates of the annual conviction rate and the cost of operating the penal system