An investor acquires the stock for $$11,000$ and the put option for $p$ dollars $($with a strike price of $$10,000$ after one year$),$ and the stock price is expected to rise to $$12,000$ with probability $q_1$ or fall to $$9,000$ with probability $q_2$ after one year $($where $q_1+q_2=1).$

$3-1)$ Find the current put option price, $p,$ and the probabilities of the stock price increasing, $q_1,$ and decreasing, $q_2,$ such that the expected returns of the stock and put option after one year are $0,$ and $b,$ respectively. $($Hint: Construct a $23$ state$-$reward matrix with the probability constraint $q_1+q_2=1$ enforced in the last row of$-$the matrix and a $2x1$ vector with $q_1$ and $q_2).$

$3-2)$ When the put option price $p$ increases, how does $b$ changes?

$3-3)$ Find the condition under which arbitrage does not occur in $(3-2).$