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 returns. 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 events are as follows:

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 programming 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 probability that you will have at least $10,000 after 3 years.