Hometown Insurance sells $10-$year annuities to retirees who are looking for stable sources of investment income. Hometown invests the annuity funds it receives in an equity index fund with annual returns that are normally distributed with a mean of $9\%$ and standard deviation of $3\mathrm{.}5\%.$ It guarantees investors a minimum annual return of $6\%$ and a maximum return $($or rate cap$)$ of $8\mathrm{.}5\%.$ This limits both the retirees$$ down$-$side risk and up$-$side return potential. Of course, Hometown makes its money on these contracts when the actual return exceeds the rate cap. Suppose a retiree invests $$50,000$ in such an annuity contract. Assume investment earnings are credited at the end of the year and are reinvested.

a$.$ What is the probability that Hometown would lose money on the contract?

b$.$

Suppose that Hometown wants to identify the minimum guaranteed annual rate of return that provides a $2\%$ chance of the company losing money on the contract. What should the minimum guaranteed annual rate of return be$?$