Problem $1(25$ points$)$Some ex$-$cons spend the rest of their lives either free, on trial, in jail, or on probation. At the start of each year, statistics show that there is $50\%$ chance that a free ex$-$con will commit a new crime and go on trial. The judge may send the ex$-$con to jail with probability $0\mathrm{.}6$ or grant probation with probability $0\mathrm{.}4.$ Once in jail, $10\%$ of ex$-$cons will be set free for good behavior. Of those who are on probation, $10\%$ commit new crimes and are arraigned for new trials, $50\%$ will go back to finish their sentence for violatin probation orders, and $10\%$ will be set free for lack of evidence. Taxpayers underwrite the cost associated with the punishment of the ex$-$felons. It is estimated that a trial will cost about $$8,000,$ an average jail sentence will cost $$25,000,$ and an average probation period will cost $$2,000.$ a$)$ Express problem as a Markov chain. Justify your answer.b$)$ Determine the expected cost per ex$-$con.c$)$ How often does an ex$-$con return to jail? Go on trial? Be set free?