Pretend that today you have the opportunity to buy a large gold nugget for $$100,000.$ You believe there's a $60\%$ chance you can sell it for $$200,000$ after five years and a $40\%$ chance to sell it for $$120,000.$ The opportunity cost of fund is $10\%$ per annum.

a$.$Do you buy the gold? Why? Show your work.

b$.$ Would buying the gold be an investment activity or a speculation activity? Why?