On Monday, a certain stock closed at $$10$ per share. Before the stock market opens on Tuesday, you expect the stock to close at $$9,$ $$10,$ or $$11$ per share, with respective probabilities $0\mathrm{.}3,0\mathrm{.}3,$ and $0\mathrm{.}4.$ Looking ahead to Wednesday, you expect the stock to close $10$ percent lower, unchanged, or $10$ percent higher than Tuesday's close, with the following probabilities.

$\backslash$table$[[,10$ Percent,,$10$ Percent$],[$Tuesday$\text{'}$s Close,Lower,Unchanged,Higher$],[$$$9,0\mathrm{.}4,0\mathrm{.}3,0\mathrm{.}3],[10,0\mathrm{.}2,0\mathrm{.}2,0\mathrm{.}6],[11,0\mathrm{.}1,0\mathrm{.}2,0\mathrm{.}7]]$

Early on Tuesday, you are directed to buy $100$ shares of the stock before Thursday. All purchases are made at the end of the day, at the known closing price for that day, so your only options are to buy at the end of Tuesday or at the end of Wednesday. You wish to determine the optimal strategy for whether to buy on Tuesday or defer the purchase until Wednesday, given the Tuesday closing price, to minimize the expected purchase price.

Develop and evaluate a decision tree.

a$-1.$ Determine the optimal strategy. If the stock closes at $$9,$ what should you do$?$

If the stock closes at $$9,$ you should

a$-2.$ Determine the expected purchase price.