$11\mathrm{.}4-2.$ Imagine that you have $$5,000$ to invest and that you will have an opportunity to invest that amount in either of two investments $(A$ or B$)$ at the beginning of each of the next $3$ years. Both investments have uncertain retums. For investment A you will either lose your money entirely or $($with higher probability$)$ get back $$10,000($a profit of $$5,000)$ at the end of the year. For investment $B$ you will get back either just your $$5,000$ or $($with low probability$)$ $$10,000$ at the end of the year. The probabilities for these cvents are as follows:

$\backslash$table$[[$Investment$,\backslash$table$[[$Amount$],[$Returned $($$$)]],$Probability$],[0,0\mathrm{.}3],[10,000,0\mathrm{.}7],[5,000,0\mathrm{.}9]]$

You are allowed to make only $($at most$)$ one investment each year, and you can invest only $$5,000$ each time. $($Any additional money accumulated is left idle.$)$

$($a$)$ Use dynamic programtning to find the investment policy that maximizes the expected amount of money you will have after $3$ years.

$($b$)$ Use dynamic programming to find the investment policy that maximizes the prohability that you will have at least $$10\mathrm{.}000$ afier $3$ years.