Problem $2.$ A basketball team trails by $4$ points with two possessions left in the game. The coach believes the team will score either $2$ points or $3$ points on each possession, with probabilities of $70\%$ and $30\%,$ respectively. The opposing team will not score any more points. Only for one time, the coach can attempt a high$-$risk $3-$point play, which has a success probability of $40\%$ but lose $2$ points with $50\%$ of chance if failed. A win has a reward of $2,$ a tie has a reward of $0\mathrm{.}4,$ and a loss is $0.$ Use dynamic programming to determine the optimal scoring strategy to maximize the team$$s expected reward. Also, show how the reward of the tie $($originally $0\mathrm{.}4)$ affects the strategy if it affects.

$*$PLEASE SHOW DETAILS OF CALCULATIONS. TO SOLVE THIS PROBELM EITHER USE DECISION TREE WITH PROBABILITY, DYNAMIC PROGRAMMING, OR THE QUEUING THEORY. BASED ON THE PROBLEMT THIS LIKELY REQUIRES DECISION TREE AND PROBABILITY CALCULATIONS.