$10.$ A business magazine rates S&P $500$ mutual fund managers based upon how often their fund beats the return on the

S&P index. Any given manager has a $50\%$ of beating the S&P index in a given year $($and $50\%$ of not beating it$),$ and

each year$$s performance is mutually independent.

$($a$)$ The magazine gives its $$Gold Star$$ rating to any manager who has beaten the S&P index in at least five of the

last six years. What is the probability of any given manager getting the $$Gold Star$$ rating?