Andrew plans to retire in $31$ years. He plans to invest part of his retirement funds in stocks, so he seeks out information on past returns. He learns that over the entire $20$th century, the real $($that is$,$ adjusted for inflation$)$ annual returns on U$.$S$.$ common stocks had mean $8\mathrm{.}7\%$ and standard deviation $20\mathrm{.}2\%.$ The distribution of annual returns on common stocks is roughly symmetric, so the mean return over even a moderate number of years is close to Normal.

$($a$)$ What is the probability $($assuming that the past pattern of variation continues$)$ that the mean annual return on common stocks over the next $31$ years will exceed $14\%?$