$1.$ A company is considering developing and marketing a new product. It is estimated to be twice as likely that the product would prove to be successful as unsuccessful. If it were successful, the expected profit would be $\backslash (\backslash$$ $1,500,000\backslash ).$ If unsuccessful, the expected loss would be $\backslash$$$3,600,000.$

a$)$ Draw a decision tree for this problem. Find the optimal strategy regarding whether to develop and market the product using the expected monetary value $($EMV$)$ rule.

b$)$ Calculate the expected value of perfect information $($EVPI$)$ for this problem.

c$)$ A marketing survey can be conducted to predict whether the product would be successful. Past experience with such surveys indicates that successful products have been predicted to be successful $80$ percent of the time, whereas unsuccessful products have been predicted to be unsuccessful $70$ percent of the time. Find the expected value of imperfect information $($EVII$)$ for the survey.