An option to buy a stock is priced at $$$$100.$ If the stock closes above $30$ on May$15,$ the option will be worth $$$$900.$ If it closes below$20,$ the option will be worth$$ nothing, and if it closes between $20$ and $30($inclusively$),$ the option will be worth $$$$100.$ A trader thinks there is a $60\%$ chance that the stock will close in the$20-30$ range, a $10\%$ chance that it will close above$30,$ and a $30\%$ chance that it will fall below $20$ on May $15.$ Complete parts$($a$)$ through$($c$).$

a$)$ How much does she expect to$$ gain?

b$)$ What is the standard deviation of her gain?

c$)$ Should she buy the stok option?