In this section, you will be expected to draw decision trees and calculate EMV for each question and answer the questions given.

$(1)$ You$$re doing a prototype for your project, but you$$re not sure whether to proceed with this prototype. If you do the prototype, it will cost you $$100,000$; and, of course, if you don$$t pursue it$,$ there will be no cost$.$ If you do the prototype, there is $30$ percent chance that the prototype might fail, and fo r that the cost impact will be $$50,000.$ However, if the prototype succeeds, the project will make $$500,000.$ If you do not do any prototype, you$$re already taking a risk, the chance of which is $80$ percent with a failure impact of $$250,000.$ But, again, without a prototype, should you succeed, the project will make the same money as mentioned before. What should you do$?$